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Mathematics 10 Learning Modules Quarter 3

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Bobbie Tolentino

This document is the introduction section of a Grade 10 mathematics learner's module developed by the Department of Education of the Philippines. It was collaboratively developed by educators from public and private schools, colleges, and universities. The introduction provides an overview of the material, noting that it is designed to support the K-12 Basic Education Program and ensure students meet expected standards. It describes the development and review process and lists the eight modules included in the material, which are aimed at developing students' critical thinking and problem-solving skills.

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D EPED C O PY 10 Mathematics Department of Education Republic of the Philippines This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at action@deped.gov.ph. We value your feedback and recommendations. Learner’s Module Unit 3 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. VISIT DEPED TAMBAYAN http://richardrrr.blogspot.com/ 1. Center of top breaking headlines and current events related to Department of Education. 2. Offers free K-12 Materials you can use and share
D EPED C O PY Mathematics – Grade 10 Learner’s Module First Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at filcols@gmail.com or (02) 439-2204, respectively. Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD Printed in the Philippines by REX Book Store Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: imcsetd@yahoo.com Development Team of the Learner’s Module Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Acelajado, PhD Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD Illustrator: Cyrell T. Navarro Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY Unit 3 Module 6: Permutations and Combinations....................................... 275 Lessons and Coverage........................................................................ 276 Module Map......................................................................................... 277 Pre-Assessment .................................................................................. 278 Learning Goals and Targets ................................................................ 282 Lesson 1: Permutations .............................................................................. 283 Activity 1.................................................................................... 283 Activity 2.................................................................................... 284 Activity 3.................................................................................... 286 Activity 4.................................................................................... 286 Activity 5.................................................................................... 295 Activity 6.................................................................................... 296 Activity 7.................................................................................... 297 Activity 8.................................................................................... 298 Activity 9.................................................................................... 299 Activity 10.................................................................................. 299 Summary/Synthesis/Generalization........................................................... 300 Table of Contents Lesson 2: Combinations..............................................................................301 Activity 1 ....................................................................................301 Activity 2 ....................................................................................303 Activity 3 ....................................................................................304 Activity 4 ....................................................................................310 Activity 5 ....................................................................................311 Activity 6 ....................................................................................311 Activity 7 ....................................................................................312 Activity 8 ....................................................................................314 Activity 9 ....................................................................................315 Activity 10 ..................................................................................315 Summary/Synthesis/Generalization ...........................................................316 Glossary of Terms.......................................................................................317 References and Website Links Used in this Module..................................317 Module 7: Probability of Compound Events ......................................319 Lessons and Coverage ........................................................................320 Module Map .........................................................................................320 Pre-Assessment...................................................................................321 Learning Goals and Targets.................................................................326 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 328 Activity 1 ....................................................................................328 Activity 2 ....................................................................................330 Activity 3 ....................................................................................332 Activity 4 ....................................................................................334 Activity 5 ....................................................................................336 Activity 6 ....................................................................................337 Activity 7 ....................................................................................338 Activity 8 ....................................................................................339 Activity 9 ....................................................................................340 Summary/Synthesis/Generalization ...........................................................340 Lesson 2: Independent and Dependent Events ..........................................341 Activity 1 ....................................................................................341 Activity 2 ....................................................................................343 Activity 3 ....................................................................................344 Activity 4 ....................................................................................344 Activity 5 ....................................................................................345 Activity 6 ....................................................................................345 Summary/Synthesis/Generalization ...........................................................345 Lesson 3: Conditional Probability..............................................................346 Activity 1 ....................................................................................346 Activity 2 ....................................................................................347 Activity 3 ....................................................................................348 Activity 4 ....................................................................................350 Activity 5 ....................................................................................351 Activity 6 ....................................................................................351 Activity 7 ....................................................................................352 Lesson 1: Union and Intersection of Events ............................................... Summary/Synthesis/Generalization........................................................... 352 Glossary of Terms ...................................................................................... 353 References and Website Links Used in this Module ................................. 354 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 275 I. INTRODUCTION Look at the pictures shown below. Have you ever wondered why some locks such as the one shown below have codes in them? Do you know why a shorter code is “weak,” while a longer code is a “strong” personal password in a computer account? Have you ever realized that there are several possible ways in doing most tasks or activities like planning a seating arrangement or predicting the possible outcomes of a race? Have you ever been aware that there are numerous possible choices in selecting from a set, like deciding which combination of dishes to serve in a catering service or deciding which dishes to order in a menu? Did you know that awareness of these can help you form conclusions and make wise decisions? Find out the answers to these questions and discover the wide applications of permutations and combinations through this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 276 II. LESSONS AND COVERAGE In this module, you will examine and determine the number of possible ways of doing certain tasks, or selecting some objects from a set. You will learn about these through the following lessons: Lesson 1 – Permutations Lesson 2 – Combinations In these lessons, you will learn to: Lesson 1  illustrate permutation of objects;  derive the formula for finding the number of permutations of n objects taken r at a time, n ≥ r ; and  solve problems involving permutations. Lesson 2  illustrate the combination of n objects;  differentiate permutation from combination of n objects taken r at a time, n ≥ r ;  derive the formula for finding the number of combinations of n objects taken r at a time; and  solve problems involving permutations and combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 277 Here is a simple map of the lessons that will be covered in this module: Finding the Permutations of n Objects Taken r at a Time Problems Involving Permutations Finding the Combinations of n Objects Taken r at a Time Problems Involving Combinations Problems Involving Permutations and Combinations Combinations Permutations COMBINATORICS All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 278 III. PRE-ASSESSMENT Part I Find out how much you already know about the topics in this module. Choose the letter that you think best answers the question. Please answer all the items. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. 1. Choosing a subset of a set is an example of ___. A. combination C. integration B. differentiation D. permutation 2. Which of the following situations or activities involve permutation? A. matching shirts and pants B. forming different triangles out of 5 points on a plane, no three of which are collinear C. assigning telephone numbers to subscribers D. forming a committee from the members of a club 3. The product of a positive integer n and all the positive integers less than it is _____. A. powers of n C. n - factors B. multiples of n D. n factorial 4. Two different arrangements of objects where some of them are identical are called _______. A. distinguishable permutations C. circular permutations B. unique combinations D. circular combinations 5. How many different 4-digit even numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition of digits is allowed? A. 1 680 B. 840 C. 420 D. 120 6. In how many ways can 8 people be seated around a circular table if two of them insist on sitting beside each other? A. 360 B. 720 C. 1440 D. 5040 7. Find the number of distinguishable permutations of the letters of the word PASS. A. 4 B. 12 C. 36 D. 144 8. Ms. Santos asked Renz to draw all the diagonals of a certain polygon on the blackboard. Renz was able to draw 27 diagonals which his teacher declared correct. What was the given polygon? A. pentagon C. nonagon B. hexagon D. decagon All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 279 9. Ms. De Leon wants to produce different sets of test questions for her essay test. If she plans to do this by putting together 3 out of 5 questions she prepared, how many different sets of questions could she construct? A. 10 B. 20 C. 60 D. 80 10. If P(9, r) = 3024, what is r ? A. 2 B. 4 C. 5 D. 6 11. In a town fiesta singing competition with 12 contestants, in how many ways can the organizer arrange the first three singers? A. 132 B. 990 C. 1320 D. 1716 12. What is P(8, 5)? A. 56 B. 336 C. 1400 D. 6720 13. If P(n, 4) = 5040, then n = ____. A. 12 B. 10 C. 9 D. 8 14. Given x = P(n, n) and y = P(n, n – 1), what can be concluded about x and y? A. x > y B. x < y C. x = y D. x = – y 15. Find the number of distinguishable permutations of the letters of the word EDUCATED. A. 1680 B. 10 080 C. 20 160 D. 40 320 16. If a combination lock must contain 5 different digits, in how many ways can a code be formed from the digits 0 to 9? A. 15 120 B. 30 240 C. 151 200 D. 1 000 000 17. In how many ways can 4 men and 3 women arrange themselves in a row for picture taking if the men and women must stand in alternate positions? A. 5040 B. 720 C. 144 D. 30 18. In a room, there are 10 chairs in a row. In how many ways can 5 students be seated in consecutive chairs? A. 720 B. 600 C. 252 D. 120 19. Which of the following situations does NOT illustrate combination? A. Selecting 2 songs from 10 choices for an audition piece B. Fixing the schedule of a group of students who must take exactly 8 subjects C. Enumerating the subsets of a set D. Identifying the lines formed by connecting some given points on a plane All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 280 20. If w = C(5, 2), x = C(5, 3), y = C(5, 4), and z = C(5, 5), and we are given 5 points on a plane of which no three are collinear, which expression gives the total number of polygons that can be drawn? A. x + y C. x + y + z B. w + x + y D. w + x + y + z 21. C(n, n) = ____. A. n B. r C. 1 D. cannot be determined 22. If C(n, r) = 35, which of the following are possible values of n and r? A. n = 6, r = 4 B. n = 7, r = 3 C. n = 8, r = 3 D. n = 9, r = 2 23. If C(n, 4) = 126, what is n? A. 11 B. 10 C. 9 D. 7 24. If C(12, r) = 792, which of the following is a possible value of r? A. 8 B. 7 C. 6 D. 4 25. A caterer offers 3 kinds of soup, 7 kinds of main dish, 4 kinds of vegetable dish, and 4 kinds of dessert. In how many possible ways can a caterer form a meal consisting of 1 soup, 2 main dishes, 1 vegetable dish, and 2 desserts? A. 140 B. 336 C. 672 D. 1512 26. In how many ways can a committee of 7 students be chosen from 9 juniors and 9 seniors if there must be 4 seniors in the committee? A. 10 584 B. 1764 C. 210 D. 84 27. Jane wants to solve a system of equations through elimination by combining any two equations. The number of equations she has is equal to the number of variables. She realizes that she has 10 possible ways to start her solution. How many equations does she have? A. 6 B. 5 C. 4 D. 3 28. There are 11 different food items in a buffet. A customer is asked to get a certain number of items. If the customer has 462 possible ways as a result, which of the following did he possibly do? A. Choose 4 out of the 11 items B. Choose 6 out of the 11 items C. Choose 8 out of the 11 items D. Choose 7 out of the 11 items All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 281 Part II Read and understand the situation below, and then answer the question or perform what is required. Suppose you had graduated from high school but did not have enough money to continue your college education. You decided to work temporarily and save for your schooling. You applied at Mr. Aquino’s restaurant and were hired. After a few days, you noticed that the restaurant business was not doing very well, and Mr. Aquino asked for your opinion. What you noticed was that there was no variety in the food being served in the restaurant. 1. Prepare a list of different choices of food that may be served (soup, meat/chicken dishes, fish, vegetables, fruits, desserts, beverages). Consider health and nutritional values. 2. What mathematical concepts are utilized in this given situation? 3. Formulate two problems involving the mathematical concepts in the situation. 4. Write the equation(s) or expressions that describe the situation. 5. Solve the equations and problems formulated. 6. Present a sample menu for the day and explain the reason for your choice. In view of the fact that there will be several possible combinations, explain also why you should prepare certain dishes more often or less frequently. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 282 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with two or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately, but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts of combinatorics, particularly permutations and combinations. Also, you should be able to use precise counting techniques in formulating conclusions and making decisions. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 283 Start Lesson 1 of this module by assessing your knowledge of the basic counting technique called the Fundamental Counting Principle. This knowledge and skill will help you understand permutations of objects. As you go through this lesson, keep in mind this important question: How does the concept of permutation help in forming conclusions and in making wise decisions? To be able to answer this, perform each activity that follows. Seek the assistance of your teacher and peers if you encounter any difficulty. Have your work checked by your teacher. A. A close friend invited Anna to her birthday party. Anna has 4 new blouses (stripes, with ruffles, long-sleeved, and sleeveless) and 3 skirts (red, pink, and black) in her closet reserved for such occasions. 1. Assuming that any skirt can be paired with any blouse, in how many ways can Anna select her outfit? List the possibilities. 2. How many blouse-and-skirt pairs are possible? 3. Show another way of finding the answer in item 1. B. Suppose you secured your bike using a combination lock. Later, you realized that you forgot the 4-digit code. You only remembered that the code contains the digits 1, 3, 4, and 7. 1. List all the possible codes out of the given digits. 2. How many possible codes are there? 3. What can you say about the list you made? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 284 Questions: a. How did you determine the different possibilities asked for in the two situations? What methods did you use? b. What did you feel when you were listing the answers? Were you able to list all the possibilities asked for? How did you ensure that your list was complete? What method(s) did you use in order to give the accurate count? Why do you think there is a need to know the number of possible ways a certain task can be done? You will find this out as you go through this lesson. Answer the following questions. 1. Ten runners join a race. In how many possible ways can they be arranged as first, second, and third placers? 2. If Jun has 12 T-shirts, 6 pairs of pants, and 3 pairs of shoes, how many possibilities can he dress himself up for the day? 3. In how many ways can Aling Rosa arrange 6 potted plants in a row? 4. How many four-digit numbers can be formed from the numbers 1, 3, 4, 6, 8, and 9 if repetition of digits is not allowed? Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 285 5. If there are 3 roads from Town A to Town B and 4 roads from Town B to Town C, in how many ways can one go from Town A to Town C and back to Town A, through Town B, without passing through the same road twice? 6. Suppose that in a certain association, there are 12 elected members of the Board of Directors. In how many ways can a president, a vice president, a secretary, and a treasurer be selected from the board? 7. In how many ways can you place 9 different books on a shelf if there is space enough for only 5 books? 8. You want to order your lunch from the school canteen, which offers student meals consisting of 1 cup of rice, 1 meat dish, and 1 vegetable dish. How many choices do you have for your meal if there are 3 choices of meat dishes and 2 choices of vegetable dishes? 9. In how many ways can 5 people arrange themselves in a row for picture taking? 10. A dress-shop owner has 8 new dresses that she wants to display in the window. If the display window has 5 mannequins, in how many ways can she dress them up? Questions: a. How did you find the answer to each of the questions? What previously learned principle did you apply? b. Show and explain how you answered each item. In the activity you have done, were you able determine the exact number of ways of doing each task or activity described? What mathematics concept or principle did you use? How was that principle applied? Some of these tasks or activities share similarities or differ from others in some sense. How do they differ? You will find out as you go through the next sections of this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 286 Take a second look at the 10 situations in Activity 2. Determine if in doing the activity or task, arrangement or order of selecting the objects is important; that is, whether a different order or arrangement means a different result. Write your answers on a manila paper and be ready to share them to the class. Questions: 1. In which situations in Activity 2 is order or arrangement of the selection important? 2. Justify your answer by giving an example for each situation. 3. In performing a certain task where order or arrangement is important, what do you call each possible arrangement? Perform the following activity using four number cards with different digits. Follow all instructions and write all your answers on a clean sheet of paper. Then, complete the table and answer the questions that follow. A. Get any two number cards (Example: 1 and 2). 1. a. Arrange the cards using 1 piece at a time. Example: 1 2 ways 2 b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made. 2. a. Arrange the cards using both pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made. B. Get any three number cards. 1. a. Arrange the cards using 1 piece at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using 1 card at a time from the 3 given cards. Activity 4: Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 287 2. a. Arrange the cards using 2 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using 2 cards at a time from the 3 given cards. 3. a. Arrange the cards using all 3 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using all the 3 cards at a time. C. Get the four number cards. 1. Repeat Steps B.1 to 3. 2. a. Arrange the objects using all 4 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made using all the 4 number cards at a time. Tabulate all results. Results: Number of Objects (n) Number of Objects Taken at a Time (r) Number of Possible Arrangements 2 1 2 2 2 3 1 3 2 3 3 4 1 4 2 4 3 4 4 Questions: a. What do you call each arrangement? b. Can you find any pattern in the results? c. Can you think of other ways of finding these answers? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 288 How did you find the preceding activities? Are you ready to learn about permutations? From the activities you have done, you recalled the Fundamental Counting Principle which is an essential tool in under- standing about arrangement, or permutations. You also identified tasks in which the arrangement or order is important. How can permutations help in solving real-life problems or in making conclusions and decisions? You will find out in the next activities. Before doing these activities, read and understand first some important notes on permutations including the examples presented. How do we find the permutations of objects? Suppose we have 6 different potted plants and we wish to arrange 4 of them in a row. In how many ways can this be done? We can determine the number of ways these plants can be arranged in a row if we arrange only 4 of them at a time. Each possible arrangement is called a permutation. The permutation of 6 potted plants taken 4 at a time is denoted by P(6, 4), 6P4 , P6,4 , or 6 4 P Similarly, if there are n objects which will be arranged r at a time, it will be denoted by P(n, r) . The permutation of n objects taken r at a time is denoted by P(n, r). In some books, it is also denoted by nPr, Pn,r , or n r P . In this learning material, we will use the first notation. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 289 How do we find the number of permutations of n objects taken r at a time? Study the following illustrations: Example 1. Given the 4-letter word READ. In how many ways can we arrange its letters, 3 at a time? Solution: There are 3 positions to be filled. We write the horizontal marks: _ _ _. On each horizontal mark, we write the number of choices we have in filling up the said position. There are 4 choices for the first position. After the first position is filled, there are 3 choices left for the second position. After the first and second positions are filled, there are 2 choices left for the third position. So the three entries on the horizontal marks would be 4 3 2 We apply the Fundamental Counting Principle and multiply: 4 ⋅ 3 ⋅ 2 = 24. Thus, there are 24 possible ways of arranging the 4 letters of READ taking 3 of them at a time. We also say the 4 letters of READ, taken 3 at a time, have 24 permutations. One of them is R-E-D. Verify this result. Let us take a second look at the example above. Remember: n = 4, r = 3. 4⋅ 3 ⋅ 2 = n(n – 1)(n – 2) Notice that the first factor is n, and the succeeding factors decrease by 1 each time. Look at the last factor: n - 2 = n – (3 – 1) = n – (r – 1) or n – r + 1. Take note also that there are r factors in all, starting with n. Example 2. In a school club, there are 5 possible choices for the president, a secretary, a treasurer, and an auditor. Assuming that each of them is qualified for any of these positions, in how many ways can the 4 officers be elected? Solution: P(5, 4) = 5 ⋅ 4 ⋅ 3 ⋅ 2 = 120 ways The number of permutations of n objects taken r at a time, P(n, r), where n ≥ r is: P(n, r) = n(n – 1)(n – 2) ⋅⋅⋅ (n – r + 1) All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 290 Note that there are r factors. From what principle was the formula derived? Can you now answer the problem earlier? Suppose we have 6 potted plants and we wish to arrange 4 of them in a row. In how many ways can this be done? What if there are 5 objects to be arranged and we would arrange all of them every time? That is, n = 5, and r = 5. Example 3. In how many ways can 5 people arrange themselves in a row for picture taking? Solution: n = 5, r = 5 P(5, 5) = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 (5 factors) = 120 possible pictures In the third example, we used all the numbers from n = 5 down to 1. Another way of writing 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 is 5! (read as 5 factorial) Similarly, 4! = 4⋅ 3⋅ 2⋅ 1 = 24 8! = 8 ⋅7⋅ 6⋅ 5⋅ 4 ⋅3 ⋅2 ⋅1 = 40 320 3! = 3⋅ 2⋅ 1 = 6 10! = 10⋅ 9 ⋅8⋅ 7⋅ 6 ⋅5⋅ 4 ⋅3⋅ 2⋅ 1= 3 628 800 1! = 1 For convenience, we define 0! = 1. Thus, in example 3, P(5, 5) = 5! = 120. Also, P(8, 8) = 8! = 40 320 P(4, 4) = 4! = 24. In addition, since P(n, r) = n(n - 1)(n - 2)⋅⋅⋅(n - r +1) = )!( )!1)(2)...(1)(( rn rnrnnnn   Then, P(n, r) = )!( ! rn n  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 291 Remember: The permutation of n objects taken r at a time is: P(n, r) = )!( ! rn n  , n ≥ r and the permutation of n objects taken all at a time is: P(n, n) = n! Notice that, in the previous examples, the objects to be arranged are all distinct. Suppose some of the objects to be arranged are not distinct, that is, some are identical. Study the following examples. Example 4. Find the number of permutations of the letters of the word EVEN. Solution: There are 4 letters of the word. Initially, suppose these letters are distinct, then the number of permutation is P(4, 4) = 4!. However, we have to take into consideration that the 2 E’s are alike. We cannot distinguish between the 2 E’s. How can we arrange the 4 letters differently if the 2 E’s could be differentiated from one another? Instead of using subscripts, let us make one E in upper case and the other in lower case (e). We will have: EVeN eVEN ENVe ENeV VNEe NEVe EVNe eVNE ENeV eNEV NEVe NeVE EeVN eEVN VEeN VeEN NEeV NeEV EeNV eENV VENe VeNE NVEe NVeE How many arrangements are there? Can you think of other possible arrangements? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 292 Now, these two E’s are the same, and so the two entries in each box are also the same. Take note that for each permutation like E-V-E-N, there are 2, or 2! ways of arranging the two E’s without changing the arrangement of the others. The duplications are eliminated by dividing 4! or 24 by the number of ways of arranging the 2 E’s which is 2! So, out of the 24 arrangements, the number of distinct or distinguishable permutations, P, would be: P = (4,4) 2! P = 4! 2 = 24 2 = 12 distinguishable permutations Example 5. Find the number of permutations of the letters of the word STATISTICS. Solution: There are 10 letters of the word. Assuming that the letters are distinct, there are P(10, 10) = 10! permutations. However, we have to take into consideration that the 3 S’s are alike, the 3 T’s are alike, and the 2 Is are also alike. The permutations of the 3 S’s is P(3, 3) = 3!. The permutations of the 3 T’s is P(3, 3) = 3! The permutation of the 2 T’s is P(2, 2) = 2! So we must divide 10! by 3! 3! 2! in order to eliminate the duplicates. Thus, P = 10! 3! 3! 2! = 50 400 permutations This leads to the next rule. The number of distinguishable permutations, P, of n objects where p objects are alike, q objects are alike, r objects are alike, and so on, is P = ! ! ! !... n p q r All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 293 Let us now consider arrangement of objects in a circle, which we call circular permutations. Example 6. a. In how many ways can 3 people be seated around a circular table? Solution: n = 3 (a) (b) Notice that the arrangement 1-2-3 in (a) above is the same as 2-3-1 and 3-1-2; meaning these 3 permutations are just one and the same. Notice also that the arrangements 1-3-2, 3-2-1, and 2-1- 3 are also just the same as seen in (b). So the circular permutations, P, of 3 objects is: P  6 3  3! 3   3 2! 3  2! 1 2 3 1 3 2 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 294 We can also look at it this way: If there are 3 seats around the circular table, we can assign one of the persons to a fixed seat and determine the number of ways that the other two can be arranged. This way, we avoid counting again an arrangement that resulted from a mere rotation. Thus, the number of circular permutations of 3 objects is (3 – 1)! or 2! and the circular permutations of n objects is (n – 1)!. b. In how many ways can 4 people be seated around a circular table? Solution: (n = 4) Let us call the 4 people A, B, C, D The arrangements are: A-B-C-D A-B-D-C A-C-B-D A-C-D-B A-D-B-C A-D-C-B B-C-D-A B-D-C-A C-B-D-A C-D-B-A D-B-C-A D-C-B-A C-D-A-B D-C-A-B B-D-A-C D-B-A-C B-C-A-D C-B-A-D D-A-B-C C-A-B-D D-A-C-B B-A-C-D C-A-D-B B-A-D-C Observe that all the arrangements falling on the same column are just the same because the 4 people are supposed to be seated around a circular table. There are 24 arrangements in the list. Again, the circular permutations, P, of 4 objects is: P = 24 4 based on the list made = 4! 4 = 4 !34  = 3! Or simply, P = (n - 1)! = (4 - 1)! = 3! = 6 The permutation of n objects arranged in a circle is P = (n - 1)! All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 295 Know more about permutations through these websites. http://mathworld.wolfram.com/circularpermutations.ht ml http://www.mathsisfun.com/combinatorics/combinatio ns-permutations.html http://users.math.yale.edu/~anechka/math107/lecture 13.pdf Your goal in this section is to apply the key concepts of permutations. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided. Solve for the unknown in each item, and then answer the questions that follow. 1. P(6, 6) = ___ 6. P(8, r) = 6 720 2. P(7, r) = 840 7. P(8, 3) = ___ 3. P(n, 3) = 60 8. P(n, 4) = 3024 4. P(n, 3) = 504 9. P(12, r) = 1320 5. P(10, 5) = ___ 10. P(13, r) = 156 Questions: a. How did you calculate the different permutations? b. What mathematics concepts or principles did you apply to solve each permutation? c. Did you find any difficulty in finding the answers? What technique or strategy can you think of to facilitate your way of solving? How did you find the preceding activity? Was it easy? I am sure it was! Find out from your peers if you had the same answers or strategies in solving. If not, discover why. In the next activity, you are going to apply the concept of permutations to solve real-life problems. Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 296 Answer each permutation problem completely. 1. A teacher wants to assign 4 different tasks to her 4 students. In how many possible ways can she do it? 2. In a certain general assembly, three major prizes are at stake. In how many ways can the first, second, and third prizes be drawn from a box containing 120 names? 3. In how many different ways can 5 bicycles be parked if there are 7 available parking spaces? 4. How many distinguishable permutations are possible with all the letters of the word ELLIPSES? 5. There are 8 basketball teams competing for the top 4 standings in order to move up to the semi-finals. Find the number of possible rankings of the four top teams. 6. In how many different ways can 12 people occupy the 12 seats in a front row of a mini-theater? 7. Find the number of different ways that a family of 6 can be seated around a circular table with 6 chairs. 8. How many 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition is allowed? 9. If there are 10 people and only 6 chairs are available, in how many ways can they be seated? 10. Find the number of distinguishable permutations of the digits of the number 348 838. Activity 6: RAFFLE DRAW All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 297 Answer each permutation problem completely. 1. In how many different ways can you arrange 8 figurines on a shelf? 2. There are 4 different Mathematics books and 5 different Science books. In how many ways can the books be arranged on a shelf if a. there are no restrictions? b. books of the same subject must be placed together? c. if they must be placed alternately? 3. Five couples want to have their pictures taken. In how many ways can they arrange themselves in a row if a. couples must stay together? b. they may stand anywhere? 4. There are 12 people in a dinner gathering. In how many ways can the host (one of the 12) arrange his guests around a dining table if a. they can sit on any of the chairs? b. 3 people insist on sitting beside each other? c. 2 people refuse to sit beside each other? 5. A teacher drew a number line on the board and named some points on it. The teacher then asked the class to list all the rays in the figure. A student answered 30 rays in all and the teacher said it was correct. How many points were named in the figure? How were the activities done so far? Were you able to answer all of the exercises? I hope you just did successfully! Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 298 In this section, permutation was discussed, including permutation with identical objects, circular permutation, and its real-life applications. You may go back to the previous sections and compare your initial ideas about the concept of permutation. Were these further clarified? Now that you have learned the basic ideas about permutations, let us deepen your understanding and study the next section. In this section, you are going to think deeper and test further your understanding of permutations. After doing the following activities, you should be able to answer the important question: How are permutations used in forming conclusions and in making wise decisions? Answer the following questions completely. 1. How do you determine if a situation or problem involves permutations? 2. Differentiate between permutation in general (n objects taken r at a time), circular permutation, and distinguishable permutation (when some objects are alike). 3. a. Find the number of permutations of n objects when arranged (n - 1) at a time for any positive integer n. b. Find the number of permutations of n objects when taken all at a time. c. Compare the two (a and b). Explain why the answers still make sense. 4. Going back to the “combination” lock problem on page 285, suppose that the lock contained the 4 digits 1, 4, 7, 9 but you totally forgot the order in which they come. What are you going to do? How can you apply your knowledge of permutations here? Elaborate. Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 299 In this section, the discussion was about your understanding of permutations and how they are illustrated and used in real life. What new realizations do you have about permutations? How would you connect these realizations to real life? Write an entry in your journal describing how much you already learned about permutations and their applications. Include also whatever points in the lesson in which you still need clarifications, and work on these with your teacher. Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which will demonstrate your understanding of permutations. Do the following tasks by group. A. Give 3 examples of problems or situations in real life that involve permutations. In each example, 1. explain the problem or situation. 2. solve the problem. 3. discuss how you can use these sample situations in your daily life, especially in formulating conclusions and/or making decisions. B. It is in international summits that major world decisions happen. Suppose that you were the overall in charge of the seating in an international convention wherein 12 country-representatives were invited. They are the prime ministers/presidents of the countries of Canada, China, France, Germany, India, Japan, Libya, Malaysia, Philippines, South Korea, USA, and United Kingdom. 1. If the seating arrangement is to be circular, how many seating arrangements are possible? 2. Create your own seat plan for these 12 leaders based on your knowledge of their backgrounds. Discuss why you arranged them that way. Activity 10: Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 300 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with two or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately , shows in- depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately , shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 In this section your task was to give examples of real-life situations where permutation is illustrated. You also formulated and solved problems related to permutation. SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about permutations and its applications in real-life situations. The lesson provided you with opportunities to identify situations that describe permutations and differentiate them from those that do not. You were also given a chance to perform practical activities for you to further understand the topic. In addition, you were given the opportunity to formulate and solve problems on permutations and apply the knowledge to formulating conclusions and making decisions. Your understanding of this lesson as well as the other Mathematics concepts previously learned will help you learn the next topic, combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 301 Start Lesson 2 of this module by assessing your knowledge of the Fundamental Counting Principle and permutations. This knowledge and skill will help you understand combinations, which will further help you in forming conclusions and in making decisions. To be able to do this, perform each activity that follows. Seek the assistance of your teacher and peers if you encounter any difficulty. Have your work checked by your teacher. Solve each problem below and answer the questions that follow. 1. If your school cafeteria offers pork, beef, chicken, and fish for main dish, chop suey, pinakbet, and black beans for vegetable dishes, banana and pineapple for dessert, and tea, juice, and softdrinks for beverage, in how many ways can you choose your meal consisting of 1 cup of rice, 1 main dish, 1 vegetable dish, 1 beverage, and 1 dessert? 2. You were tasked to take charge of the auditions for the female parts of a stage play. In how many possible ways can you form your cast of 5 female members if there were 15 hopefuls? 3. If ice cream is served in a cone, in how many ways can Abby choose her three-flavor ice cream scoop if there are 6 available flavors? 4. If each Automated Teller Machine card of a certain bank has to have 4 different digits in its passcode, how many different possible passcodes can there be? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 302 5. How many possible permutations are there in the letters of the word PHILIPPINES? 6. In how many ways can a group of 6 people be seated around a table if 2 of them insist on sitting beside each other? 7. In how many ways can 4 students be seated in a classroom if there are 7 available seats? 8. In how many ways can 5 English books and 4 Mathematics books be placed on a shelf if books of the same subject are to be together? 9. A family catering business prides itself with its offerings of delicious meals and other food requirements. If you were one of the staff or key persons in this business, how can you apply your knowledge of permutation and combination to further improve your business? 10. Due to the huge population of Mapayapa High School, one of the problems encountered is the big crowd through the gates during the early morning and at dismissal time in the afternoon. If you were one of the administrators of the school, what step can you suggest to remedy this problem? a. How did you find the number of ways asked for in each item? What mathematics concepts or principles did you apply? How did you apply these concepts or principles? b. Which situations above illustrate permutations? Which do not? Why? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 303 Were you able to find what was required in each problem? Were you able to recall and apply the different mathematics concepts or principles in solving each problem? Why do you think it is important to be able to perform such mathematical tasks? You will find this out as you go through this lesson. Study the tasks or activities below, and then answer the questions that follow. 1. Choosing 5 questions to answer out of 10 questions in a test 2. Opening a combination lock 3. Winning in a contest 4. Selecting 7 people to form a Student Affairs Committee 5. Forming triangles from 6 distinct points in which no 3 points are collinear 6. Assigning seats to guests at dinner 7. Drawing a set of 6 numbers in a lottery containing numbers 1 to 45 8. Entering the PIN (Personal Identification Number) of your ATM card 9. Selecting 3 posters to hang out of 6 different posters 10. Listing the elements of subsets of a given set Questions: a. In which tasks/activities above is order or arrangement important? Give an example to illustrate each answer. b. In which tasks/activities is order not important? Give an example to illustrate each answer. In the activity you have just done, were you able to identify situations that involve permutations and those that do not? The latter are called combinations and you will learn more about them in the next activities. Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 304 Perform the following activity using any 5 different fruits (mango, guava, banana, pomelo, avocado, or any fruits available in your place). Follow all instructions and write all your answers on a clean sheet of paper. Then, complete the table and answer the questions that follow. A. Get 2 fruits (ex. mango and banana). 1. a. Select a fruit 1 piece at a time. Do all possible selections. b. Illustrate or describe each selection you made. Example: mango banana  2 ways c. Count the number of different selections you have made. 2. a. Select the fruits 2 pieces at a time. b. Illustrate or describe each selection. c. Count the number of different selections you have made. B. Get 3 fruits. 1. a. Select a fruit 1 piece at a time. Do all possible selections. b. Illustrate or describe each selection. c. Count the number of different selections you have made when using 1 object at a time from the 3 given fruits. 2. a. Select 2 fruits at a time. b. Illustrate or describe each selection. c. Count the number of selections you have made, using 2 objects at a time from the 3 given fruits. 3. a. Select 3 fruits at a time. b. Illustrate or describe each selection. c. Count the number of selections you have made, using all the 3 fruits at a time. C. Get 4 fruits. 1. Repeat Steps B.1 to 3. 2. a. Continue the process until you select 4 fruits at a time. b. Illustrate or describe each selection. c. Count the number of different selections you have made using all the 4 fruits at a time. D. Repeat the same procedure for 5 fruits. Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 305 Results: Number of Objects (n) Number of Objects Taken at a Time (r) Number of Possible Selections 2 1 2 2 3 1 3 2 3 3 4 1 4 2 4 3 4 4 5 1 5 2 5 3 5 4 5 5 Questions: 1. Did it matter in what order you selected the objects? 2. Give an example to justify your answer in (1). 3. What do you call each unique selection? 4. Can you find any pattern in the results? 5. Can you think of other ways of finding these answers? How did you find the preceding activities? Are you ready to learn about combinations? From the activities you have done, you identified tasks in which the arrangement or order is important (permutations) and those in which order is not (combinations). You recalled the Fundamental Counting Principle as well as permutation which are essential concepts in understanding combinations. In the next sections of this learning module, keep in mind the important question: How can the concept of combinations help in solving real-life problems or in formulating conclusions and making wise decisions? You will find out when you do the next activities. Before doing these activities, read and understand first some important notes on combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 306 Suppose you were assigned by your teacher to be the leader of your group for your project. You were given the freedom to choose 4 of your classmates to be your group mates. If you choose Aira, Belle, Charlie, and Dave, does it make any difference if you choose instead Charlie, Aira, Dave, and Belle? Of course not, because the list refers to the same people. Each selection that you could possibly make is called a combination. On the other hand, if you choose Aira, Belle, Dave, and Ellen, now that is another combination, and it is different from the first combination cited. Remember: If there is a set S with n elements, and if r is a nonnegative integer less than or equal to n, then each subset of S containing r distinct elements is called a combination of S. The number of combinations of n objects taken r at a time is denoted by C(n, r) , nCr , n rC or       r n . In this learning material, we will use the first notation. How do we find the number of combinations of n objects taken r at a time? Suppose now, that you are asked to form different triangles out of 4 points plotted, say, A, B, C, and D, of which no three are collinear. We can see that ABC is the same as BCA and CBA. In the same manner, BCD is the same as CBD and DBC. This is another illustration of combination. The different triangles that can be formed are ABC, ABD, BCD, and CDA. Thus, there are 4 combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 307 How can we find the number of combinations more systematically? Consider this: If order of the letters is important, then we have the following possibilities: ABC ABD BCD CDA ACB ADB BDC CAD BCA BDA CDB DAC BAC BAD CBD DCA CAB DBA DBC ADC CBA DAB DCB ACD The number of different orders of 4 vertices taken 3 at a time is given by P(4, 3) = 4! (4 3)! = 24. There are 24 possibilities. Since you learned in Geometry that we can name a triangle using its three vertices in any order, then if we look more closely, we can see that all the triangles in the same column are identical. Thus, the actual number of combinations is C(4, 3) = 24 6 or ( )4, 6 3P or ( 3 4, ! 3)P . (Equation 1) Notice that 6 or 3! is the number of ways of arranging 3 objects taken all at a time. We divided by 3! to eliminate duplicates. Note : There are 4 objects (A,B,C,D)  n = 4 They are selected 3 at a time  r = 3. And so equation (1) becomes C(n,r) = n,P( r) r! . Since P(n, r) = )!( ! rn n  , then C(n, r) = n,P( r) r! = ! )!( ! r rn n  = )!(! ! rnr n  . All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 308 Remember: The combination of n objects taken r at a time is: C(n, r) = ! ! ! n r (n r ) , n ≥ r ≥ 0 Example 1. In how many ways can a committee consisting of 4 members be formed from 8 people? Solution 1: (Using the formula) n = 8, r = 4 C(n,r) = ! ! ( )! n r n r = 8! 4! (8 4)! = 2 8 7 6 5 4 3 2 1 4 3 2 1 4 3 2 1               = 7 2  5 after common factors are cancelled = 70 ways Solution 2: Based on the discussion on the previous page, C(n,r) = ( , ) ! P n r r =   8 7 6 5 4! =       2 8 7 6 5 4 3 2 1 = 7  2 5 after common factors are cancelled = 70 ways There are r = 4 factors Permutation of n objects taken r at a time Permutation of r objects All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 309 Example 2. How many polygons can be possibly formed from 6 distinct points on a plane, no three of which are collinear? Solution: The polygon may have 3, 4, 5, or 6 vertices. Thus, the number N of possible polygons is: N = C(6, 3) + C(6, 4) + C(6, 5) + C(6, 6) = 6 5 4 3!   + 6 5 4 3 4!    + 6 5 4 3 2 5!     + 6 5 4 3 2 1 6!      = 6 5 4 3 2 1     + 6 5 4 3 4 3 2 1       + 6 5 4 3 2 5 4 3 2 1         + 6 5 4 3 2 1 6 5 4 3 2 1           = 5 ∙ 4 + 5 ∙ 3 + 6 + 1 = 20 + 15 + 7 N = 42 possible polygons. Verify this answer by using the formula for combinations. Read more about combinations in these websites. www.beatthegmat.com/mba/2013/09/27/does-order- matter-combinations-and-non-combinations http://www.mathsisfun.com/combinatorics/combinatio ns-permutations.html http://users.math.yale.edu/~anechka/math107/lecture 13.pdf http://voices.yahoo.com/the-importance- permutations-combinations-in-10262.html All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 310 Your goal in this section is to apply the key concepts of combinations. Use the mathematical ideas and the examples presented in the preceding section to answer the given activities. Study the following situations. Then, answer the questions that follow. 1. Determining the top three winners in a Science Quiz Bee 2. Forming lines from six given points with no three of which are collinear 3. Forming triangles from 7 given points with no three of which are collinear 4. Four people posing for pictures 5. Assembling a jigsaw puzzle 6. Choosing 2 household chores to do before dinner 7. Selecting 5 basketball players out of 10 team members for the different positions 8. Choosing three of your classmates to attend your party 9. Picking 6 balls from a basket of 12 balls 10. Forming a committee of 5 members from 20 people Questions: a. In the items above, identify which situations illustrate permutation and which illustrate combination. b. How did you differentiate the situations that involve permutation from those that involve combination? Were you able to differentiate the tasks/situations that involve permutation from those that involve combination? The next activity will provide you with more exercises to improve your calculating skills. Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 311 Find the unknown in each item. 1. C(8, 3) = ___ 6. C(10, r) = 120 2. C(n, 4) = 15 7. C(n, 2) = 78 3. C(8, r) = 28 8. C(11, r) = 165 4. C(9, 9) = ___ 9. C(8, 6) = ___ 5. C(n, 3) = 35 10. C(14,10) = ___ How did you find the activity? Was it easy? I guess it was! Go on to the next activity to apply your knowledge of combinations in real-life situations. Solve the following problems completely. (Choose a partner.) 1. If there are 12 teams in a basketball tournament and each team must play every other team in the eliminations, how many elimination games will there be? 2. If there are 7 distinct points on a plane with no three of which are collinear, how many different polygons can be possibly formed? 3. How many different sets of 5 cards each can be formed from a standard deck of 52 cards? 4. In a 10-item Mathematics problem-solving test, how many ways can you select 5 problems to solve? 5. In problem number 4, how many ways can you select the 5 questions if you are required to answer question number 10? 6. In how many ways can a committee of 5 be formed from 5 juniors and 7 seniors if the committee must have 3 seniors? 7. From a population of 50 households, in how many ways can a researcher select a sample with a size of 10? 8. A box contains 5 red balls, 7 green balls, and 6 yellow balls. In how many ways can 6 balls be chosen if there should be 2 balls of each color? Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 312 9. From 7 Biology books and 6 Chemistry books, in how many ways can one select 2 Biology and 2 Chemistry books to buy if all the said books are equally necessary? 10. Mrs. Rivera’s business is gown rental and sale. She decided one day that she would display her 10 newest gowns in her shop’s window to attract customers. If she only had 5 mannequins and planned to change the set of gowns every 2 days, how many days will have to pass before she runs out of a new set to display? Were you able to do the activity successfully? I am sure you did! Go on to the next activity to see if you can solve the following set of permutation and combination problems. Solve the following permutation and combination problems. 1. In how many ways can you arrange 5 Mathematics books, 4 Science books, and 3 English books on a shelf such that books of the same subject are kept together? 2. In how many ways can 6 students be seated in a row of 6 seats if 2 of the students insist on sitting beside each other? 3. In a gathering, the host makes sure that each guest shakes hands with everyone else. If there are 25 guests, how many handshakes will be done? Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 313 4. Suppose you are the owner of a sari-sari store and you want to put 12 pieces of canned goods in a row on the shelf. If there are 3 identical cans of meat loaf, 4 identical cans of tomato sauce, 2 identical cans of sardines, and 3 identical cans of corned beef, in how many different ways can you display these goods? 5. A soloist is auditioning for a musical play. If she is required to sing any three of the 7 prepared songs, in how many ways can she make her choice? 6. In a dance contest, each participating group must perform 3 kinds of dance. If there are 4 choices for ballroom dance, 8 choices for foreign dance, and 5 choices for hip-hop, in how many possible ways can a dance group select their piece? 7. If 3 marbles are picked randomly from a jar containing 6 red marbles and 8 green marbles, in how many possible ways can it happen that at least 2 of the marbles picked are green? 8. There are 10 identified points on a number line. How many possible rays can be drawn using the given points? 9. You are transferring to a new house. You have a collection of books but you cannot take them all with you. In how many ways can you select 7 books out of 10, and then arrange these books on a shelf if there is space enough for only 5 books? 10. At Enzo’s Pizza Parlor, there are seven different toppings, where a customer can order any number of these toppings. If you dine at the said pizza parlor, with how many possible toppings can you actually order your pizza? How did you find all the activities you have done? I hope you have answered the exercises correctly. In this section, combination was discussed, including its real-life applications. You were also given the opportunity to differentiate combination from permutation. Now that you have learned the basic ideas about combination, let us deepen your understanding and study the next section. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 314 In this section you are going to think more deeply and test further your understanding of combinations. After doing the following activities, you should be able to answer the important question: How are combinations used in forming conclusions and making wise decisions? Answer the following questions completely. 1. How do you determine if a situation involves combinations? 2. To find the total number of polygons that can be formed from 7 points on a plane with no three of which are collinear, Joy answered: C(7, 3) = !3!4 !7 = !3!4 !4567  = 123 567   = 35 different polygons Is Joy correct? Justify your answer. 3. (a) In how many ways can the 12 members of the Board of Directors (BOD) be chosen from 12 parent-nominees and 7 teacher-nominees if there must be 8 parents in the BOD? (b) After the 12 members are chosen, in how many ways can they elect among themselves the 7 top positions (president, vice president, and others)? 4. DAMATH is a board game that incorporates mathematical skills in the Filipino game Dama. In a school DAMATH tournament, there are 28 participants who are divided into 7 groups. Each participant plays against each member of his group in the eliminations. The winner in each group advances to the semi-finals where they again compete with each other. The five players with the most number of wins proceed to the final round and play against each other. Assume that there are no ties. a. What is the total number of games to be played in the eliminations? b. How many matches will be played in the final round? Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 315 c. In how many possible ways can the top five players in the semi- finals come up? d. In how many possible ways can the 1st, 2nd, and 3rd placer be declared in the final round? e. How many matches will be played altogether? In this section, the discussion was about your understanding of combinations and how they are illustrated and used in real life. What new learnings do you have about combinations? How can these learnings be applied in real life? Write an entry in your journal describing how much you have learned about combinations, and how these can be applied to real life. Add also the parts of the lesson that you still find confusing, if any. Work on these difficulties with your teacher. Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which requires that you demonstrate your understanding of combinations. Answer the following problems completely. Give 3 examples of situations in real life that illustrate combinations. In each situation, 1. formulate a problem. 2. solve the problem. 3. explain how this particular problem may help you in formulating conclusions and/or making decisions. Activity 10: Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 316 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes. 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately , shows in-depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately , shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about combinations and their applications in real life. Through the lesson, you were able to identify situations that describe combinations and differentiate them from those that do not. You were also given the opportunity to perform practical activities to further understand the topic, formulate related real-life problems, and solve these problems. You also applied your knowledge to formulating conclusions and making wise decisions. Your understanding of this lesson, combined with other previously learned Mathematics concepts will help you understand the next lesson, probability. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 317 GLOSSARY OF TERMS Circular permutation – the different possible arrangements of objects in a circle. The number of permutations, P, of n objects around a circle is given by P = (n – 1)!. Combinations – the number of ways of selecting from a set when the order is not important. The number of combinations of n objects taken r at a time is given by C(n, r) = !)!( ! rrn n  , n ≥ r. Distinguishable permutations – refers to the permutations of a set of objects where some of them are alike. The number of distinguishable permutations of n objects when p are alike, q are alike, r are alike, and so on, is given by P = !...!! ! rqp n . Fundamental Counting Principle – states that if activity A can be done in n1 ways, activity B can be done in n2 ways, activity C in n3 ways, and so on, then activities A, B, and C can be done simultaneously in  321 nnn ways. Permutations – refers to the different possible arrangements of a set of objects. The number of permutations of n objects taken r at a time is: P(n, r) = )!( ! rn n  , n ≥ r. n-Factorial – the product of the positive integer n and all the positive integers less than n. n! = n(n – 1)(n – 2) … (3)(2)(1). REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bennett, J. & Chard, D., et al. (2005). Pre-Algebra. Texas: Holt, Rinehart and Winston. Bhowal, M. & Barua, P. (2008). Statistics: 2nd ed. New Delhi: Kamal Jagasia. Leithold, L. (2002). College Algebra and Trigonometry. Singapore: Pearson Education Asia Pte. Ltd. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 318 McCune, S. (2010). Statistics. New York: The Mc-Graw-Hill Companies, Inc. Kelly, W. and Donnelly, R. (2009). The Humungous Book of Statistics Problems. New York: Penguin Group (USA), Inc. Spiegel, M.R. & Stephens, L.J. (2008). Schaum’s Outline of Theory and Problems of Statistics 4th ed. New York: The Mc-Graw-Hill Companies, Inc. Website Links as References and Sources of Learning Activities: BBC News About International Summits. Jonathan Powell. May 29, 2012. www.bbc.co.uk/news/magazine-18237721 Accessed on Feb. 27, 2014 Circular Permutation. Weisstein, Eric W. From Mathworld - A Wolfram Web Resource. http://mathworld.wolfram.com/circularpermutations.html Accessed on Feb. 26, 2014 Combinations and Permutations http://www.mathsisfun.com/combinatorics/combinations-permutations.html Accessed on Feb. 26, 2014 Does Order Matter- Combinations and Non-Combinations. Brent Hanneson. Sept. 27, 2013. www. beatthegmat.com/mba/2013/09/27/does-order-matter- combinations-and-non-combinations-partiii. Accessed on Feb. 27, 2014 Mathematics in the Real World. http://users.math.yale.edu/~anechka/math107/lecture13.pdf. Accessed on Feb. 25, 2014 The Fundamental Counting Principle and Permutations. http://www.classzone.com/eservices/home/pdf/student/LA212AAD.pdf. Accessed on Feb. 25, 2014 The Importance of Permutations and Combinations in Modern Society. Valerie Hansen.Nov.7, 2005. http://voices.yahoo.com/the-importance- permutations-combinations-in-10262.html?cat=41 Accessed on Feb. 26, 2014 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 319 I. INTRODUCTION It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. The most important questions of life are indeed, for the most part, really only problems of probability. Pierre Simon de Laplace Maybe, you are familiar with games of chance such as lotto. You have also learned that the likelihood of winning in any game of chance can be estimated using probability. However, do you not know that the theory of probability is not only for games of chance? Probability is very useful in many practical and important fields in science, engineering, economics, and social sciences. Why do you think is the study of probability so important to the advancement of science and technology? You will learn the reasons when you study this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 320 II. LESSONS AND COVERAGE Lesson 1 – Probability of Compound Events  Union and Intersection of Events  Mutually Exclusive and Not Mutually Exclusive Events Lesson 2 – Probability of Independent Events Lesson 3 – Conditional Probability In these lessons, you will learn to: Lesson 1  illustrate events, and union and intersection of events  illustrate the probability of a union of two events and intersection of events  illustrate and find probabilities of mutually exclusive events Lesson 2  illustrate independent and dependent events  find probabilities of independent and dependent events Lesson 3  identify conditional probabilities  solve problems on conditional probabilities Probability of Compound Events Probability of Compound Events Union and Intersection of Events Probability of Independent Events Conditional Probability Mutually Exclusive and Not Mutually Exclusive Events All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 321 III. PRE-ASSESSMENT Part I: Multiple Choice Answer all of the following questions the best that you can. As much as possible, provide your own solution. Take note of the items that you were not able to answer correctly and find out the right answer as you go through this module. 1. Brian likes to wear colored shirts. He has 10 shirts in the closet. Three of these are blue, four are in different shades of red, and the rest are of mixed or different colors. What is the probability that he will wear a blue or a red shirt? A. 7 4 10 10  B. 3 4 10 10  C. 3 7 10 10  D. 7 4 10 10  2. The spinner on the right is spun. What is the probability of a spin that results in an even number or a number less than 4? A. 1 4 B. 3 8 C. 4 8 D. 5 8 3. Jody has four cans of juice – one can of orange, one of pineapple, one of calamansi, and one of guyabano. She chooses three of these cans to take to school. If she chooses calamansi, what is the probability she also chooses pineapple? A. 7 8 B. 3 4 C. 2 3 D. 3 8 4. A man tosses a fair coin eight times, and observes whether the toss yields a head (H) or a tail (T). Which of the following sequences of outcomes yields a head (H) on his next toss? (I) T T T T T T T T (II) H H T H T T H H A. I C. Neither I nor II B. II D. Either I or II All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 322 5. A baby has 5 blocks in a box. One block is red, one is yellow, one is green, one is blue, and one is black. The baby pulls out a block, looks at it, and puts it back in the box. If he does this 4 times before he gets bored and crawls away, what is the probability that the 4 blocks selected are all of the same color? A. 4 5 5 B. 4 1 5 C. 4 4 5 D. 4 2 5 6. A box contains 4 red balls and 6 blue balls. A second box contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each box. The probability that both balls are of the same color is 0.44. How many blue balls are there in the second box? A. 4 B. 20 C. 24 D. 44 7. A family has two children. Suppose that the birth of each child is an independent event and that it is equally likely to be a boy or a girl. Let C denote the event that the family has one boy and one girl. Let D denote the event that the family has at most one girl. Which of the following must be true about events C and D? A. C and D are independent events. B. C occurs given that D does not occur. C. C and D are not independent events. D. C and D are mutually exclusive events. 8. A nationwide survey revealed that 42% of the population likes eating pizza. If two people are randomly selected from the population, what is the probability that the first person likes eating pizza while the second one does not? A. 0.42 + (1 − 0.42) C. 1 - 0.42 B. 2 (1 − 0.42) D. 0.42 (1 − 0.42) 9. A married couple agreed to continue bearing a new child until they get two boys, but not more than 4 children. Assuming that each time that a child is born, the probability that it is a boy is 0.5, independent from all other times. Find the probability that the couple has at least two girls. A. 1 2 B. 5 16 C. 5 8 D. 4 15 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 323 10.Some street foods were sampled and tested for the presence of disease-causing bacteria or harmful chemicals. A random sample of 200 street foods of various types according to how they are prepared was examined. The table below shows the results: Type of Food Number of Food with Bacteria Only Number of Food with Harmful Chemicals Only Number of Food with Both Bacteria and Harmful Chemicals Total Fried 35 15 18 68 Boiled 46 14 32 92 Grilled 24 8 8 40 Total 105 37 58 200 a.) What is the probability that a street food selected at random is fried? A. 68 200 B. 35 105 C. 50 142 D. 68 142 b.) What is the probability that a food selected at random is both grilled and contained harmful chemical? A. 40 200 B. 58 105 C. 16 142 D. 16 200 c.) What is the probability that a randomly selected food has both bacteria and harmful chemicals? A. 40 200 B. 58 200 C. 16 142 D. 8 58 11.A survey of a group’s sports viewing habits over the last year revealed the following information: i. 28% watched soccer ii. 29% watched basketball iii. 19% watched tennis iv. 14% watched soccer and basketball v. 12% watched basketball and tennis vi. 10% watched soccer and tennis vii. 8% watched all three sports. What percent of the group watched none of the three sports last year? A. 24 B. 36 C. 41 D. 52 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 324 12.The probability that a visit to the school clinic is neither due to dental reasons nor medical reasons is 35%. Of those coming to the clinic, 30% are due to medical reasons and 40% are due to dental reasons. What is the probability that a visit to the school clinic is due to both dental and medical reasons? A. 0.05 B. 0.12 C. 0.18 D. 0.25 13.A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and of these 312 men, 102 died from causes related to heart disease. Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. A. 102 625 B. 108 625 C. 312 625 D. 414 625 14.There are four batteries, and one of them is defective. Two are to be selected at random for use on a particular day. Find the probability that the second battery selected is not defective, given that the first was not defective. A. 2 3 B. 1 4 C. 1 3 D. 1 2 For numbers 15 to 16: A sample of 150 plastic pipes were selected and subjected to shock resistance and scratch resistance tests. The results are summarized in the table below. Scratch Resistance Shock Resistance High Low High 125 12 Low 7 6 15.A pipe is selected at random. What is the probability that it has high shock resistance given that it has high scratch resistance? A. 125 132 B. 125 137 C. 137 150 D. 132 150 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 325 16.What is the probability that it has high scratch resistance and high shock resistance? A. 125 150 B. 125 137 C. 137 150 D. 132 150 17.An insurance agent offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are 1 4 , 1 3 , and 5 12 , respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage. A. 7 9 B. 1 2 C. 47 144 D. 9 144 18.There are 24 dolphins in an ocean park. The caretaker tags 6 of them with small chips and returns them to the ocean park. The next month, he randomly selects five dolphins from the ocean park. a.) Find the probability that exactly two of the selected dolphins are tagged. A. 2 3 5 6 18 24 C C C B. 3 2 5 6 18 24 C C C C. 3 2 5 6 18 24 P P P D. 2 3 5 6 18 24 P P P b.) What were some of your assumptions in part (a) of this item? Justify your answer. Part II: Problem Solving Read and understand the situations below and solve the problem. Show your complete solution. 19.Varsity Try-Out. Suppose you are applying as a tennis varsity player of a team. To be accepted, you need to play with the team’s good player (G) and top player (T) in three games and win against both G and T in two successive games. You must choose one of the two schedules: playing G, T, G or T, G, T. Which one should you choose? Why? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 326 20.Food Preference Survey. A school conducts a survey of their students’ food preference during lunch and gets the following information: i. All students prefer to have at least one viand. ii. 70% of the students prefer to have more than one viand iii. 20% of the students prefer to have fish as viand. iv. Of those students who prefer to have more than one viand, 15% prefer fish as viand. Find the probability that a randomly selected student prefers to have exactly one viand and that is fish. Show your solution. IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of the key concepts of probability of compound events, mutually exclusive events, independent events, and of conditional probability. With these expected knowledge and skills, you should be able to use probability in formulating conclusions and in making decisions. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 327 Scoring Rubrics for Problem Solving 1 2 3 4 Problem Solving No strategy is chosen, or a strategy chosen will not lead to a solution. Little or no evidence of en- gagement in the task present A partially correct strategy is chosen, or a correct strategy for solving only part of the task is chosen. Evidence of drawing on some relevant previous knowledge is present, showing some relevant engagement in the task. A correct strategy is chosen based on the mathematical situation in the task. Planning or monitoring of strategy is evident. Evidence of solidifying prior knowledge and applying it in problem- solving is present. Note: At this level a student must achieve a correct answer. An efficient strategy is chosen and progress towards a solution is evaluated. Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered. Evidence of analyzing the situation in mathematical terms and extending prior knowledge is present. Note: At this level a student must achieve a correct answer. Reasoning Arguments are made with mathematical basis. No correct reasoning nor justification for reasoning is present. Arguments are made with some mathematical basis. Some correct reasoning or justification for reasoning is present. Arguments are constructed with adequate math- ematical basis. A systematic approach and/or justification of correct reasoning is present. Deductive arguments are used to justify decisions and may result in formal proofs. Evidence is used to justify and support decisions made and conclusions reached. Communication No awareness of audience or purpose is communicated. No formal mathematical terms or symbolic notations are evi- dent. Some communication of an approach is evident through verbal/written accounts and explanations. An attempt is made to use formal math language. One formal math term or symbolic notation is evident. Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response. Formal math language is used to share and clarify ideas. At least two formal math terms or symbolic notations are evident, in any combination. Communication of argument is supported by mathematical properties. Formal math language and symbolic notation is used to consolidate math thinking and to communicate ideas. At least one of the math terms or symbolic notations is beyond grade level. Connection No connections are made or connections are mathematically or contextually irrelevant. A mathematical connection is attempted but is partially incorrect or lacks contextual relevance. A mathematical connection is made. Proper contexts are identified that link both the mathematics concepts and the situation presented in the task. Mathematical connections are used to extend the solution to other mathematics problems or to gain a deeper understanding of the mathematics concepts. Some examples may include one or more of the following: testing and accepting or rejecting a hypothesis as an explanation of a phenomenon. Representation No attempt is made to construct a mathematical representation. An attempt is made to construct a mathematical representation to re- cord and communicate problem solving but is not accurate. An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions. An appropriate math- ematical representation is constructed to analyze relationships, extend thinking, and clarify or interpret a phenomenon. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 328 Begin Lesson 1 of this module by assessing your knowledge and skills of the different mathematics concepts related to counting techniques and probability of simple events as well as concepts of sets you previously studied. These knowledge and skills are important in understanding the probability of compound events. As you go through this lesson, think of this question, Why do you think is the study of probability important in making decisions in real life? Consider the situation below. Use your knowledge on probability in answering the questions that follow. 1. A die is rolled once. Find the probability of obtaining a. a 5. b. a 6. c. an odd number. 2. A box contains 3 red balls, 5 yellow balls, and 2 blue balls. If a ball is picked at random from the box, what is the probability that a ball picked is a a. yellow ball? b. red ball? The above activity helped you recall your knowledge of probability of simple events. If you roll a die, the number that would come up could be 1, 2, 3, 4, 5, or 6. When the die is rolled, it is equally likely to land on one face as on any other. Therefore, the probability of getting a “5” is one out of 6. In symbol, we use P(getting a 5) = 1 6 . Always remember that 1 6 is the probability that any of the faces shows up. Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 329 Activities such as rolling a die, tossing a coin, or randomly choosing a ball from a box which could be repeated over and over again and which have well- defined results are called experiments. The results of an experiment are called outcomes. The set of all outcomes in an experiment is called a sample space. An event is a subset of the sample space. Simple Events: Consider rolling a die. a. “Getting a number 5” is called a simple event. b. “Getting a 6” is also a simple event. What about the event of “getting an odd number”? Probability of Simple Events: If each of the outcomes in a sample space is equally likely to occur, then the probability of an event E, denoted as P(E) is given by   number�of�ways�the�event�can�occur number�of�possible�outcomes P E  or   number�of�outcomes�in�the�event number�of�outcomes�in�the�sample�space P E  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 330 Consider the situation below. Use the tree diagram given below in answering the questions that follow. A school canteen serves lunch for students. A set of menu consists of 1 type of rice, 1 type of viand, and 1 type of drink. The tree diagram below shows the possible menu combinations. Rice Viand Drink pineapple juice chicken adobo orange juice Fried rice pineapple juice pinakbet orange juice pineapple juice chicken adobo orange juice Steamed rice pinakbet pineapple juice orange juice 1. Give the sample space of combination of rice, viand, and drink. How many possible outcomes are there? 2. List the outcomes of selecting a lunch with pineapple juice. 3. How many outcomes are there for selecting any lunch with pineapple juice? 4. How many outcomes are there for selecting a lunch with steamed rice and with pineapple juice? 5. How many outcomes are there for selecting a lunch with chicken adobo and a pineapple juice? 6. How many outcomes are there for selecting a lunch with pinakbet and an orange juice? A student taking lunch in the canteen is selected at random. 7. What is the probability that the student chose pineapple juice as a drink? 8. What is the probability that the student chose steamed rice and pineapple juice? 9. What is the probability that the student chose chicken adobo and orange juice? 10. What is the probability that the student chose pinakbet and pineapple juice? Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 331 Reflect: a. What does the tree diagram tell you? b. How did you determine the sample space? c. Differentiate an outcome from a sample space. Give another example of an outcome. d. Aside from the tree diagram, how else can you find the total number of possible outcomes? e. Describe the outcome in this situation as compared to the events that you studied in grade 8. From the above activity, you were able to recognize that the events in the given situation are not simple events. This is because in finding the sample, you need to find first the sample space using the fundamental counting principle. The events mentioned here are called compound events. Typically, when the sample space is finite, any subset of the sample space is an event. Any subset of the sample space is an event. Since all events are sets, they are usually written as sets (e.g., {1, 2, 3}). Compound Events: Events which consist of more than one outcome are called compound events. A compound event consists of two or more simple events. Example: Finding the probability of “getting a 6 and a 1” when two dice are rolled is an event consisting of (1, 6), (6, 1) as outcomes. The first die falls in 6 different ways and the second die also falls in 6 different ways. Thus, using the fundamental counting principle, the number of outcomes in the sample space is 66 or 36. The outcomes in the sample space are: {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3),…,(6, 5), (6, 6)}. Take note that “getting a 6 and a 1” when two dice are rolled is an event consisting of {(1, 6), (6, 1)} as outcomes. This is a compound event. It is often useful to use a Venn diagram to visualize the probabilities of events. To understand more about the probability of the union and intersection of events, you may proceed to Activity 3. The next activity will help you understand the concept of intersection and union of events. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 332 The extracurricular activities in which the senior class at Kananga National High School participate are shown in the Venn diagram below. 1. How many students are in the senior class? 2. How many students participate in athletics? 3. If a student is randomly chosen, what is the probability that the student participates in athletics or drama? 4. If a student is randomly chosen, what is the probability that the student participates only in drama and band? Reflect: a. How were you able to find the total number of students in the senior class? b. How does the concept of set help you in finding the intersection and union of two or more events? c. What are some notations that are used in your study of sets in grade 7 that you can still recall? Do you think these are needed in the study of probability of compound events? Activity 3: Extra-curricular activities participated by senior students All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 333 Illustrative Example: The Venn diagram below shows the probabilities of grade 10 students joining either soccer (S) or basketball (B). Use the Venn diagram to find the probabilities. a. P(B) b. P(S) c. P(BS) d. P(BS) e. P(B’S’) To further understand the above problem, read the discussion of the solution. Actually, the diagram does not show or represent the entire sample space for B and S. What is shown are the probabilities. a. To find the P(B), we will add the probability that only B occurs to the probability that B and S occur to get 0.4 + 0.3 = 0.7. So, P(B) = 0.7. b. Similarly, P(S) = 0.2 + 0.3 = 0.5 c. Now, P(BS) is the value 0.3 in the overlapping region d. P(BS) = 0.4 + 0.3 +0.2 = 0.9. e. P(B’S’) = 1 - P(BS) = 0.1 Complement of an Event The complement of an event is the set of all outcomes that are NOT in the event. This means that if the probability of an event, A, is P(A), then the probability that the event would not occur (also called the complementary event) is 1 – P(A), denoted by P(A’). Thus, P(A’) = 1 – P(A). So the complement of an event E is the set of all the outcomes which are not in E. And together the event and its complement make all possible outcomes. Consider item e on this page. B S All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 334 P(B’S’) can be determined by finding the part of the diagram where everything outside of B overlaps with everything outside of S. It is the region outside of both circles and that probability is 0.1. Another way to think of this is P(BS)’ or 1 - P(BS). The next activity will help you understand the concepts of events which are mutually exclusive and which are not mutually exclusive. As you answer the next activity, try to figure out events which are mutually exclusive and which are not. Consider the situations below and answer the questions that follow. 1. A bowl contains 15 chips numbered 1 to 15. If a chip is drawn randomly from the bowl, what is the probability that it is a. 7 or 15? b. 5 or a number divisible by 3? c. even or divisible by 3? d. a number divisible by 3 or divisible by 4? 2. Dario puts 44 marbles in a box in which 14 are red, 12 are blue, and 18 are yellow. If Dario picks one marble at random, what is the probability that he selects a red marble or a yellow marble? 3. Out of 5200 households surveyed, 2107 had a dog, 807 had a cat, and 303 had both a dog and a cat. What is the probability that a randomly selected household has a dog or a cat? Reflect: a. How did you answer each question? b. What do you notice about the events in each question? (e.g., 1.c as compared to 1. d, question 2 as compared to question 3). c. Draw a Venn diagram showing the sample space for numbers 3 and 4. What do you notice about the Venn diagrams? Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 335 To help you understand the concepts of mutually exclusive and non- mutually exclusive events, read the discussion below. The events in the above activity may be either mutually exclusive or not mutually exclusive. Events that cannot occur at the same time are called mutually exclusive events. Consider the Venn diagram below. What do you notice about the events A and B? These two events are mutually exclusive. In problem 1b of the preceding activity, the event of getting a 5 and the event of getting a number divisible by 3 from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} are mutually exclusive events. If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. In symbols, P(A or B) = P(A) + P(B) On the other hand, the event of getting a number divisible by 3 or the event of getting a number divisible by 4 in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} are not mutually exclusive events. Observe that a subset of numbers divisible by 3 also contains an element which is a subset of the numbers divisible by 4. The Venn diagram below shows events A and B which are not mutually exclusive because A and B intersect. Note that there are outcomes that are common to A and B which is the intersection of A and B. P (A or B) A B All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 336 If two events, A and B, are not mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities decreased by the probability of both occurring. In symbols, P(A or B)= P(A)+ P(B) – P(A and B). This section requires you to use the mathematical ideas you learned from the activity and from the discussion. Answer the problems in the following activities in different ways when possible. Consider the situation below and answer the questions that follow. 1. A restaurant serves a bowl of candies to their customers. The bowl of candies Gabriel receives has 10 chocolate candies, 8 coffee candies, and 12 caramel candies. After Gabriel chooses a candy, he eats it. Find the probability of getting candies with the indicated flavors. a. P (chocolate or coffee) c. P (coffee or caramel) b. P (caramel or not coffee) d. P (chocolate or not caramel) 2. Rhian likes to wear colored shirts. She has 15 shirts in the closet. Five of these are blue, four are in different shades of red, and the rest are of different colors. What is the probability that she will wear a blue or a red shirt? 3. Mark has pairs of pants in three different colors, blue, black, and brown. He has 5 colored shirts: a white, a red, a yellow, a blue, and a mixed-colored shirt. What is the probability that Mark wears a black pair of pants and a red shirt on a given day? 4. A motorcycle licence plate has 2 letters and 3 numbers. What is the probability that a motorcycle has a licence plate containing a double letter and an even number? Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 337 Did you answer all the questions correctly? To help you understand more about mutually exclusive events, you may proceed to Activity 6. Consider each problem below. Draw a Venn diagram for each. Determine whether the events are mutually exclusive or not mutually exclusive. Then, find the probability. 1. Mario has 45 red chips, 12 blue chips, and 24 white chips. What is the probability that Mario randomly selects a red chip or a white chip? 2. Of 240 students, 176 are on the honor roll, 48 are members of the varsity team, and 36 are in the honor roll and are also members of the varsity team. What is the probability that a randomly selected student is on the honor roll or is a member of the varsity team? 3. Ruby’s dog has 8 puppies. The puppies include white females, 3 mixed-color females, 1 white male, and 2 mixed-color males. Ruby wants to keep one puppy. What is the probability that she randomly chooses a puppy that is female and white? 4. Carl’s basketball shooting records indicate that for any frame, the probability that he will score in a two-point shoot is 30%, a three-point shoot, 45%, and neither, 25%. What is the probability that Cindy will score either in a two-point shoot or in a three-point shoot? In previous lessons, you learned about counting techniques and you were able to differentiate permutation from combination. In the next activity, observe how the concepts of permutation and combination are used in solving probability problems. Activity 6: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 338 Consider the situation below and answer the questions that follow. There are a total of 48 students in Grade 10 Charity. Twenty are boys and 28 are girls. 1. If a teacher randomly selects a student to represent the class in a school meeting, what is the probability that a a. boy is chosen? b. girl is chosen? 2. If a committee of 3 students is formed, what is the probability that a. all are girls? b. two are boys and one is a girl? 3. Suppose that a team of 3 students is formed such that it is composed of a team leader, a secretary, and a spokesperson. What is the probability that a team formed is composed of a girl secretary? Reflect: a. How did you answer each question? b. In finding the probability of each event above, what concepts are needed? c. Differentiate the event required in question 1 as compared to questions 2 and 3. d. Compare the events in questions 2 and 3. What necessary knowledge and skills did you need to get the correct answer? How did you compute for the probability of an event in each case? Notice that the above problems involved concepts of combination and permutation in determining the sample space and in determining the events. Do you remember your previous lesson on combination and permutation? How useful are your knowledge and skills on permutation and combination in solving problems on probability? Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 339 This time you need to reflect on and further find out your understanding of compound events and mutually and not mutually exclusive events. You may also apply your knowledge and skills on the Fundamental Counting Principle, combinations, and permutations in determining the possible outcomes in the sample space. Answer the following questions 1. How does a simple event differ from a compound event? 2. Differentiate mutually exclusive events from non-mutually exclusive events. 3. Suppose there are three events A, B, and C that are not mutually exclusive. List all the probabilities you would need to consider in order to calculate P(A or B or C). Then, write the formula you would use to calculate the probability. 4. Explain why subtraction is used when finding the probability of two events that are not mutually exclusive. What new realizations do you have about probability of compound events? How would you connect this topic to other concepts that you have previously learned? How would you use this in real life? Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 340 This section gives an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of probability of compound events, mutually exclusive events, and not mutually exclusive events. Answer the following questions. Write a report of your answers using a minimum of 120 words. Be ready to present your answers in the class. 1. Describe a situation in your life that involves events which are mutually exclusive or not mutually exclusive. Explain why the events are mutually exclusive or not mutually exclusive. 2. Think about your daily experience. How is probability utilized in newspapers, television shows, and radio programs that interest you? What are your general impressions of the ways in which probability is used in the print media and entertainment industry? SUMMARY/SYNTHESIS/GENERALIZATION In this lesson, you learned that an event is any collection of outcomes of an experiment. Typically, when the sample space is finite, any subset of the sample space is an event. Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events. You also learned that to each event, a probability is assigned. The probability of a compound event can be calculated if its outcomes are equally likely. Events can be mutually exclusive or not mutually exclusive. If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. In symbols, P(A or B) = P(A) + P(B). On the other hand, if two events, A and B, are not mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities decreased by the probability of both of them occurring. In symbols, P (A or B) = P (A) + P (B) – P (A and B). Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 341 In Lesson 1 of this module, you learned about the basic concepts of the probability of compound events. You will formally learn, for example, why the outcome of the flip of a fair coin is independent of the flips that came before it. Consider the situations below and answer the questions that follow. Situation 1: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. A ball is drawn at random and the color is noted and then put back inside the box. Then, another ball is drawn at random. Find the probability that: a. both are blue. b. the first is red and the second is yellow. Situation 2: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. Suppose that two balls are drawn one after the other without putting back the first ball. Find the probability that: a. the first is red and the second is blue. b. both balls are yellow. Reflect: a. Compare the process of getting the probabilities in each of the situations above? b. In situation 1, is the probability of obtaining the second ball affected by the first ball? What about in situation 2? c. What conclusion can you make about events happening in the given situations above? How are these events different? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 342 You have just completed Activity 1 on understanding dependent and independent events. Continue by considering the discussion that follows. Please read thoroughly and if needed, you may go back to the situations given in Activity 1 of this lesson. Independent and Dependent Events In situation 1, the probability of getting a blue ball in the second draw is not affected by the probability of drawing a red ball on the first draw, since the first ball is put back inside the box prior to the second draw. Thus, the two events are independent of each other. Two events are independent if the outcome of one event does not affect the outcome of the other event. Example: When a coin is tossed and a die is rolled, the event that a coin shows up head and the event that a die shows up a 5 are independent events. Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other. If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B) = P(A) • P(B) When the outcome of one event affects the outcome of another event, they are dependent events. In situation 2, if the ball was not placed back in the box, then drawing the two balls would have been dependent events. In this case, the event of drawing a yellow ball on the second draw is dependent on the event of drawing a yellow ball on the first draw. Example: A box contains 7 white marbles and 7 red marbles. What is the probability of drawing 2 white marbles and 1 red marble in succession without replacement? On the first draw, the probability of getting a white marble is 7 14 . On the second draw, the probability of getting a white marble is 6 13 . Then on the third draw, the probability of getting a red marble is 7 12 . So, P(1 white, 1 white, 1 red) = 7 6 7 7 14 13 12 52    All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 343 If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) = P(A) • P(B following A) This section requires you to use the mathematical ideas you learned from the activity and from the discussion. Answer the problems in the following activities in different ways when possible. Consider the situation below and answer the questions that follow. 1. A bag contains 6 black marbles, 9 blue marbles, 4 yellow marbles, and 2 green marbles. A marble is randomly selected, replaced, and a second marble is randomly selected. Find the probability of selecting a black marble, then a yellow marble. 2. A box of chocolates contains 10 milk chocolates, 8 dark chocolates, and 6 white chocolates. Hanissa randomly chooses a chocolate, eats it, and then randomly chooses another chocolate. What is the probability that Hanissa chose a milk chocolate, and then, a white chocolate? 3. A rental agency has 12 white cars, 8 gray cars, 6 red cars, and 3 green cars for rent. Mr. Escobar rents a car, returns it because the radio is broken, and gets another car. What is the probability that Mr. Escobar is given a green car and then a gray car? Did you answer all the questions correctly? If not, you may go back to the discussion. Otherwise you may proceed to Activity 3. Now, using your own words, differentiate independent events from dependent events. Then, you may answer the questions in the next activity. Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 344 Determine whether the events are independent or dependent. Then, find the probability. 1. A toy box contains 12 toys, 8 stuffed animals, and 3 board games. Maria randomly chooses 2 toys for the child she is babysitting to play with. What is the probability that she chose 2 stuffed animals as the first two choices? 2. A basket contains 6 apples, 5 bananas, 4 oranges, and 5 guavas. Dominic randomly chooses one piece of fruit, eats it, and chooses another piece of fruit. What is the probability that he chose a banana and then an apple? 3. Nick has 4 black pens, 3 blue pens, and 2 red pens in his school bag. Nick randomly picks two pens out of his school bag. What is the probability that Nick chose two blue pens, if he replaced the first pen back in his pocket before choosing a second pen? This time you need to reflect on and further find out your understanding of dependent and independent events. Consider the situation below and answer the questions that follow. 1. A bag of jelly beans contains 10 red, 6 green, 7 yellow, and 5 orange jelly beans. What is the probability of randomly choosing a red jelly bean, replacing it, randomly choosing another red jelly bean, replacing it, and then randomly choosing an orange jelly bean? 2. Rene and Cris went to a grocery store to buy drinks. They chose from 10 different brands of juice drinks, 6 different brands of carbonated drinks, and 3 different brands of mineral water. What is the probability that Rene and Cris both chose juice drinks, if Rene randomly chose first and liked the first brand he picked up? Activity 4: Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 345 Answer the following questions. 1. What makes two events independent? 2. Differentiate a dependent event from an independent event. Reflect: a. What new realizations do you have about the probability of a dependent event? b. How would you connect this topic to other concepts that you have previously learned? c. How would you use these concepts in real life? This section is an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of probability of compound events, dependent events, and independent events. 1. Describe a situation in your life that involves dependent and independent events. Explain why the events are dependent or independent. 2. Formulate your own problems involving independent and dependent events. SUMMARY/SYNTHESIS/GENERALIZATION Two events are independent if the occurrence of one of the events gives no information about whether or not the other event will occur; that is, the events have no influence on each other. If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B) = P(A) • P(B). When the outcome of one event affects the outcome of another event, they are dependent events. If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) = P(A) • P(B following A). Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 346 Conditional probability plays a key role in many practical applications of probability. In these applications, important conditional probabilities are often drastically affected by seemingly small changes in the basic information from which the probabilities are derived. In this lesson, we will focus on finding a particular kind of probability called a conditional probability. To understand conditional probability, you may begin by answering Activity 1. Consider the situation and answer the questions that follow. Mario bought four different batteries. Of these four, one is defective. Two are to be selected at random for use on a particular day. 1. Draw a tree diagram associated with the experiment of selecting two batteries from among four, in which one is defective. 2. List the sample space. How many outcomes are there? 3. Find the probability that the second battery selected is not defective. 4. What if you find the probability that the second battery selected is not defective, given that the first was not defective? Reflect: a. How did you answer question number 3? How is the condition different from that of question number 4? b. How did you find the probability that the second battery selected was not defective, given that the first was not defective? In Activity 1, you notice that a condition was given when you were asked to find the probability of an event. How does the given condition affect the probability of an event? Activity 1 shows an example of probability involving conditions which we refer to as conditional probability. To understand conditional probability further, you may proceed to Activity 2. Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 347 Consider the situation and answer the questions that follow. A proficiency examination for certain technical skills was given to 100 employees of a manufacturing firm. The breakdown of the examination results of the employees are shown in the table below. Male (M) Female (F) Total Pass (P) 24 36 60 Fail (Q ) 16 24 40 40 60 100 Suppose that an employee is selected at random from among the 100 employees who took the examination. 1. Are the events P and M independent? Explain. 2. Are the events P and F independent? Explain. 3. Find the probability that the employee passed the exam, given that he was a male. 4. Find the probability that the employee was a male, given that a passing grade was received. 5. Find the probability that the employee was a female, given that a passing grade was received. In many situations, once more information become available, we are able to revise our estimates for the probability of further outcomes or events happening. In this lesson, we are interested in answering this type of question: how the information “an event B has occurred" affects the probability that “event A occurs." The usual notation for "event A occurs given that event B has occurred" is "A|B" (A given B). The symbol | is a vertical line and does not imply division. P(A|B) denotes the probability that event A will occur given that event B has occurred already. We define conditional probability as follows: For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by ( ) ( | ) ( ) P A B P A B P B   Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 348 When two events, A and B, are dependent, the probability of both events occurring is P(A and B) = P(B) P(A|B). Also, P(A and B) = P(A) P(B|A). Sample Problem: A mathematics teacher gave her class two tests. Twenty- five percent of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test? Solution: This problem involves a conditional probability since it asks for the probability that the second test was passed given that the first test was passed.       P First and Second P Second|First P First  0.25 0.42  25 42   0.60 or 60% Consider the table below showing A as the age group under 30 years old who purchase 2 different brands of shoes. Age Group Brand X Brand Y Total A (under 30 years old) 6% 34% 40% A’ (30 years and older) 9% 51% 60% Total 15% 85% 100% 1. What is the probability that a person chosen at random purchases Brand X? 2. What is the probability that a person chosen at random is under 30 years old? 3. What is the probability that a person chosen at random purchases Brand X and is under 30 years old? 4. What is the probability that a person chosen at random purchases Brand X and he or she is under 30 years old? Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 349 Question for discussion: Does the occurrence of event A give information about the probability of event X? Why or Why not? To understand further conditional probability, study the solution to the above problem. Solution: We use a Venn diagram to show the relationships of the events. 1. P(X) = 0.15 2. P(A) = 0.40 3. P X A( )= 0.06 4.     ( ) | P P X P X A A A   . . 0 06 0 40  .0 15 Take note that from 1) and 4), P(X) and P(X|A) are both equal to 0.15. Notice that the occurrence of event A gives no information about the probability of event X. The events X and A are independent events. Two events A and B are said to be independent if either: i. P(A | B) = P(A), i.e., P(B | A) = P(B), or equivalently, ii. P(A ∩ B) = P(A) P(B). Probabilities are usually very sensitive to an information given as a condition. Sometimes, however, a probability does not change when a condition is supplied. If the extra information provided by knowing that an event B has occurred does not change the probability of A, that is, if, ( | ) ( )P A B P A then events A and B are said to be independent. Since     ( ) | P P A P A B B B   Notice that two events A and B are said to be independent if P(A B) = P(A) P(B). This is equivalent to stating that P(A|B) = P(A) and P(B|A) = P(B) if these probabilities exist. 0.06 0.51 0.09 S X A 0.34 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 350 Sometimes a conditional probability is known, and we want to find the probability of an intersection. By rearranging the terms in the definition of conditional probability and considering the definition of independence, we obtain the Multiplicative Rule which is P(A ∩ B) = P(A)P(B). Let us look at some other problems in which you are asked to find a conditional probability in Activity 4. This section requires you to use the mathematical ideas you learned from the previous activities and from the discussion. Answer the problems in the following activities in different ways when possible. Answer the following questions. 1. Given that P(S) = 0.3, P(S|Q) = 0.4, and P(Q) = 0.5, find the following probabilities: a. P(SQ) b. P(Q|S) c. P(S’|Q) d. P(S|Q’) 2. Assume S and Q are independent events with P(S) = 0.2 and P(Q) = 0.3. Let T be the event that at least one of S or Q occurs, and let R be the event that exactly one of S or Q occurs. Find the following probabilities a. P(T) b. P(R) c. P(S|R) d. P(R|S). e. Determine whether S and R are independent. Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 351 Reflect: a. What do you notice about the conditional probability of independent events? b. How about the conditional probability of dependent events? 1. A family has two children. What is the probability that the younger child is a girl, given that at least one of the children is a girl? 2. At a basketball game, 80% of the fans cheered for team A. In the same crowd, 20% of the fans were waving banners and cheering for team A. What is the probability that a fan waved a banner given that the fan cheered for team A? This time you need to reflect on and further find out your understanding of dependent and independent events. For numbers 1 to 3, consider the Venn diagram on the right. 1. What does the Venn diagram illustrate? 2. Using the Venn diagram, how do you find P(B|A)? 3. Write a situation that can be illustrated by the Venn diagram. Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 352 This section provides an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of conditional probability. Make a research report. Choose your own topic of study or choose from any of the four recommended topics given below. Focus on the question that follows: How can I use statistics and probability to help others make informed decisions regarding my chosen topic? Recommended Topics: 1. Driving and cell phone use 2. Diet and health 3. Professional athletics 4. Costs associated with a college education SUMMARY/SYNTHESIS/GENERALIZATION For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by     ( ) | P P A P A B B B   In word problems, conditional probabilities can usually be recognized by words like “given”, “if,” or “among” (e.g., in the context of samples). There are, however, no hard rules, and you have to read the problem carefully and pay attention to the entire context of the problem to determine whether the given probability represents an ordinary probability (e.g., P(AB)) or a conditional probability (e.g., P(A|B) or P(B|A)). Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 353 GLOSSARY OF TERMS Complement of an Event – a set of all outcomes that are NOT in the event. If A is the event, the complement of the event A is denoted by A’ Compound Events – a composition of two or more simple events Conditional Probability - The conditional probability of an event B given A is the probability that the event B will occur given that an event A has already occurred. This probability is written as P(B|A) and read as the probability of B given A. In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is, P(B). Dependent Events – Two events are dependent if the occurrence of one event does affect the occurrence of the other (e.g., random selection without replacement). Events – a set of possible outcomes resulting from a particular experiment. For example, a possible event when a single six-sided die is rolled is {5, 6}, that is, the roll could be a 5 or a 6. In general, an event is any subset of a sample space (including the possibility of an empty set). Independent Events – events in which the probability of any one event occurring is unaffected by the occurrence or non-occurrence of any of the other events. Formally, A and B are independent if and only if P(A|B) = P(A). Intersection of Events – a set that contains all of the elements that are in both events. The intersection of events A and B is written as A B . Mutually Exclusive Events – events that have no outcomes in common. This also means that if two or more events are mutually exclusive, they cannot happen at the same time. This is also referred to as disjoint events. Union of Events – a set that contains all of the elements that are in at least one of the two events. The union is written as .A B Venn Diagram – a diagram that uses circles to represent sets, in which the relations between the sets are indicated by the arrangement of the circles. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
D EPED C O PY 354 REFERENCES AND WEBSITE LINKS USED IN THIS MODULE References: Canonigo, A. (2000). Historical, Philosophical and Psychological Considerations in Probability Theory Intersection: Official Journal of Philippine Council of Mathematics Teacher Educators (MATHED), Inc. Lee-Chua, Editor. MATHED, Inc: Quezon City. 1(2), 22-32. Carpenter, T. Corbitt, M., Kepner, H., Lindquist, M. & Reys, R. (1981). “What are the chances of your students knowing probability?” The Mathematics Teacher 74(5), 342-344. Website Links as Reference and Sources of Learning Activities: Conditional Probability and Independence. Tutorials for Finite Math. http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/frames6_5.html Accessed on 3 March 2014. Conditional Probability. Wolfram MathWorld. http://mathworld.wolfram.com/ConditionalProbability.html Accessed on 4 March 2014. Conditional Probability: Definitions and non-trivial examples. Cut the Knot. http://www.cut-the-knot.org/Probability/ConditionalProbability.shtmlAccessed on 6 March 2014. Conditional Probability.http://www.stat.yale.edu/Courses/1997- 98/101/condprob.htm Accessed on 6 March 2014 Independent Events. Math Goodies. http://www.mathgoodies.com/lessons/vol6/independent_events.html Accessed on 6 March 2014. Mutually Exclusive Events. Math Is Fun. http://www.mathsisfun.com/data/probability-events-mutually-exclusive.html Accessed on 6 March 2014 The Monty Hall Dilemma. Cut the Knot .http://www.cut-the-knot.org/hall.shtml Accessed on 6 March 2014 The Monty Hall Problem.http://montyhallproblem.com/ Accessed on 6 March 2014 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

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Mathematics 10 Learning Modules Quarter 3

  • 1. D EPED C O PY 10 Mathematics Department of Education Republic of the Philippines This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at action@deped.gov.ph. We value your feedback and recommendations. Learner’s Module Unit 3 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. VISIT DEPED TAMBAYAN http://richardrrr.blogspot.com/ 1. Center of top breaking headlines and current events related to Department of Education. 2. Offers free K-12 Materials you can use and share
  • 2. D EPED C O PY Mathematics – Grade 10 Learner’s Module First Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at filcols@gmail.com or (02) 439-2204, respectively. Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD Printed in the Philippines by REX Book Store Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: imcsetd@yahoo.com Development Team of the Learner’s Module Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Acelajado, PhD Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD Illustrator: Cyrell T. Navarro Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 3. D EPED C O PY Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 4. D EPED C O PY Unit 3 Module 6: Permutations and Combinations....................................... 275 Lessons and Coverage........................................................................ 276 Module Map......................................................................................... 277 Pre-Assessment .................................................................................. 278 Learning Goals and Targets ................................................................ 282 Lesson 1: Permutations .............................................................................. 283 Activity 1.................................................................................... 283 Activity 2.................................................................................... 284 Activity 3.................................................................................... 286 Activity 4.................................................................................... 286 Activity 5.................................................................................... 295 Activity 6.................................................................................... 296 Activity 7.................................................................................... 297 Activity 8.................................................................................... 298 Activity 9.................................................................................... 299 Activity 10.................................................................................. 299 Summary/Synthesis/Generalization........................................................... 300 Table of Contents Lesson 2: Combinations..............................................................................301 Activity 1 ....................................................................................301 Activity 2 ....................................................................................303 Activity 3 ....................................................................................304 Activity 4 ....................................................................................310 Activity 5 ....................................................................................311 Activity 6 ....................................................................................311 Activity 7 ....................................................................................312 Activity 8 ....................................................................................314 Activity 9 ....................................................................................315 Activity 10 ..................................................................................315 Summary/Synthesis/Generalization ...........................................................316 Glossary of Terms.......................................................................................317 References and Website Links Used in this Module..................................317 Module 7: Probability of Compound Events ......................................319 Lessons and Coverage ........................................................................320 Module Map .........................................................................................320 Pre-Assessment...................................................................................321 Learning Goals and Targets.................................................................326 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 5. D EPED C O PY 328 Activity 1 ....................................................................................328 Activity 2 ....................................................................................330 Activity 3 ....................................................................................332 Activity 4 ....................................................................................334 Activity 5 ....................................................................................336 Activity 6 ....................................................................................337 Activity 7 ....................................................................................338 Activity 8 ....................................................................................339 Activity 9 ....................................................................................340 Summary/Synthesis/Generalization ...........................................................340 Lesson 2: Independent and Dependent Events ..........................................341 Activity 1 ....................................................................................341 Activity 2 ....................................................................................343 Activity 3 ....................................................................................344 Activity 4 ....................................................................................344 Activity 5 ....................................................................................345 Activity 6 ....................................................................................345 Summary/Synthesis/Generalization ...........................................................345 Lesson 3: Conditional Probability..............................................................346 Activity 1 ....................................................................................346 Activity 2 ....................................................................................347 Activity 3 ....................................................................................348 Activity 4 ....................................................................................350 Activity 5 ....................................................................................351 Activity 6 ....................................................................................351 Activity 7 ....................................................................................352 Lesson 1: Union and Intersection of Events ............................................... Summary/Synthesis/Generalization........................................................... 352 Glossary of Terms ...................................................................................... 353 References and Website Links Used in this Module ................................. 354 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 6. D EPED C O PY 275 I. INTRODUCTION Look at the pictures shown below. Have you ever wondered why some locks such as the one shown below have codes in them? Do you know why a shorter code is “weak,” while a longer code is a “strong” personal password in a computer account? Have you ever realized that there are several possible ways in doing most tasks or activities like planning a seating arrangement or predicting the possible outcomes of a race? Have you ever been aware that there are numerous possible choices in selecting from a set, like deciding which combination of dishes to serve in a catering service or deciding which dishes to order in a menu? Did you know that awareness of these can help you form conclusions and make wise decisions? Find out the answers to these questions and discover the wide applications of permutations and combinations through this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 7. D EPED C O PY 276 II. LESSONS AND COVERAGE In this module, you will examine and determine the number of possible ways of doing certain tasks, or selecting some objects from a set. You will learn about these through the following lessons: Lesson 1 – Permutations Lesson 2 – Combinations In these lessons, you will learn to: Lesson 1  illustrate permutation of objects;  derive the formula for finding the number of permutations of n objects taken r at a time, n ≥ r ; and  solve problems involving permutations. Lesson 2  illustrate the combination of n objects;  differentiate permutation from combination of n objects taken r at a time, n ≥ r ;  derive the formula for finding the number of combinations of n objects taken r at a time; and  solve problems involving permutations and combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 8. D EPED C O PY 277 Here is a simple map of the lessons that will be covered in this module: Finding the Permutations of n Objects Taken r at a Time Problems Involving Permutations Finding the Combinations of n Objects Taken r at a Time Problems Involving Combinations Problems Involving Permutations and Combinations Combinations Permutations COMBINATORICS All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 9. D EPED C O PY 278 III. PRE-ASSESSMENT Part I Find out how much you already know about the topics in this module. Choose the letter that you think best answers the question. Please answer all the items. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. 1. Choosing a subset of a set is an example of ___. A. combination C. integration B. differentiation D. permutation 2. Which of the following situations or activities involve permutation? A. matching shirts and pants B. forming different triangles out of 5 points on a plane, no three of which are collinear C. assigning telephone numbers to subscribers D. forming a committee from the members of a club 3. The product of a positive integer n and all the positive integers less than it is _____. A. powers of n C. n - factors B. multiples of n D. n factorial 4. Two different arrangements of objects where some of them are identical are called _______. A. distinguishable permutations C. circular permutations B. unique combinations D. circular combinations 5. How many different 4-digit even numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition of digits is allowed? A. 1 680 B. 840 C. 420 D. 120 6. In how many ways can 8 people be seated around a circular table if two of them insist on sitting beside each other? A. 360 B. 720 C. 1440 D. 5040 7. Find the number of distinguishable permutations of the letters of the word PASS. A. 4 B. 12 C. 36 D. 144 8. Ms. Santos asked Renz to draw all the diagonals of a certain polygon on the blackboard. Renz was able to draw 27 diagonals which his teacher declared correct. What was the given polygon? A. pentagon C. nonagon B. hexagon D. decagon All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 10. D EPED C O PY 279 9. Ms. De Leon wants to produce different sets of test questions for her essay test. If she plans to do this by putting together 3 out of 5 questions she prepared, how many different sets of questions could she construct? A. 10 B. 20 C. 60 D. 80 10. If P(9, r) = 3024, what is r ? A. 2 B. 4 C. 5 D. 6 11. In a town fiesta singing competition with 12 contestants, in how many ways can the organizer arrange the first three singers? A. 132 B. 990 C. 1320 D. 1716 12. What is P(8, 5)? A. 56 B. 336 C. 1400 D. 6720 13. If P(n, 4) = 5040, then n = ____. A. 12 B. 10 C. 9 D. 8 14. Given x = P(n, n) and y = P(n, n – 1), what can be concluded about x and y? A. x > y B. x < y C. x = y D. x = – y 15. Find the number of distinguishable permutations of the letters of the word EDUCATED. A. 1680 B. 10 080 C. 20 160 D. 40 320 16. If a combination lock must contain 5 different digits, in how many ways can a code be formed from the digits 0 to 9? A. 15 120 B. 30 240 C. 151 200 D. 1 000 000 17. In how many ways can 4 men and 3 women arrange themselves in a row for picture taking if the men and women must stand in alternate positions? A. 5040 B. 720 C. 144 D. 30 18. In a room, there are 10 chairs in a row. In how many ways can 5 students be seated in consecutive chairs? A. 720 B. 600 C. 252 D. 120 19. Which of the following situations does NOT illustrate combination? A. Selecting 2 songs from 10 choices for an audition piece B. Fixing the schedule of a group of students who must take exactly 8 subjects C. Enumerating the subsets of a set D. Identifying the lines formed by connecting some given points on a plane All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 11. D EPED C O PY 280 20. If w = C(5, 2), x = C(5, 3), y = C(5, 4), and z = C(5, 5), and we are given 5 points on a plane of which no three are collinear, which expression gives the total number of polygons that can be drawn? A. x + y C. x + y + z B. w + x + y D. w + x + y + z 21. C(n, n) = ____. A. n B. r C. 1 D. cannot be determined 22. If C(n, r) = 35, which of the following are possible values of n and r? A. n = 6, r = 4 B. n = 7, r = 3 C. n = 8, r = 3 D. n = 9, r = 2 23. If C(n, 4) = 126, what is n? A. 11 B. 10 C. 9 D. 7 24. If C(12, r) = 792, which of the following is a possible value of r? A. 8 B. 7 C. 6 D. 4 25. A caterer offers 3 kinds of soup, 7 kinds of main dish, 4 kinds of vegetable dish, and 4 kinds of dessert. In how many possible ways can a caterer form a meal consisting of 1 soup, 2 main dishes, 1 vegetable dish, and 2 desserts? A. 140 B. 336 C. 672 D. 1512 26. In how many ways can a committee of 7 students be chosen from 9 juniors and 9 seniors if there must be 4 seniors in the committee? A. 10 584 B. 1764 C. 210 D. 84 27. Jane wants to solve a system of equations through elimination by combining any two equations. The number of equations she has is equal to the number of variables. She realizes that she has 10 possible ways to start her solution. How many equations does she have? A. 6 B. 5 C. 4 D. 3 28. There are 11 different food items in a buffet. A customer is asked to get a certain number of items. If the customer has 462 possible ways as a result, which of the following did he possibly do? A. Choose 4 out of the 11 items B. Choose 6 out of the 11 items C. Choose 8 out of the 11 items D. Choose 7 out of the 11 items All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 12. D EPED C O PY 281 Part II Read and understand the situation below, and then answer the question or perform what is required. Suppose you had graduated from high school but did not have enough money to continue your college education. You decided to work temporarily and save for your schooling. You applied at Mr. Aquino’s restaurant and were hired. After a few days, you noticed that the restaurant business was not doing very well, and Mr. Aquino asked for your opinion. What you noticed was that there was no variety in the food being served in the restaurant. 1. Prepare a list of different choices of food that may be served (soup, meat/chicken dishes, fish, vegetables, fruits, desserts, beverages). Consider health and nutritional values. 2. What mathematical concepts are utilized in this given situation? 3. Formulate two problems involving the mathematical concepts in the situation. 4. Write the equation(s) or expressions that describe the situation. 5. Solve the equations and problems formulated. 6. Present a sample menu for the day and explain the reason for your choice. In view of the fact that there will be several possible combinations, explain also why you should prepare certain dishes more often or less frequently. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 13. D EPED C O PY 282 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with two or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately, but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts of combinatorics, particularly permutations and combinations. Also, you should be able to use precise counting techniques in formulating conclusions and making decisions. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 14. D EPED C O PY 283 Start Lesson 1 of this module by assessing your knowledge of the basic counting technique called the Fundamental Counting Principle. This knowledge and skill will help you understand permutations of objects. As you go through this lesson, keep in mind this important question: How does the concept of permutation help in forming conclusions and in making wise decisions? To be able to answer this, perform each activity that follows. Seek the assistance of your teacher and peers if you encounter any difficulty. Have your work checked by your teacher. A. A close friend invited Anna to her birthday party. Anna has 4 new blouses (stripes, with ruffles, long-sleeved, and sleeveless) and 3 skirts (red, pink, and black) in her closet reserved for such occasions. 1. Assuming that any skirt can be paired with any blouse, in how many ways can Anna select her outfit? List the possibilities. 2. How many blouse-and-skirt pairs are possible? 3. Show another way of finding the answer in item 1. B. Suppose you secured your bike using a combination lock. Later, you realized that you forgot the 4-digit code. You only remembered that the code contains the digits 1, 3, 4, and 7. 1. List all the possible codes out of the given digits. 2. How many possible codes are there? 3. What can you say about the list you made? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 15. D EPED C O PY 284 Questions: a. How did you determine the different possibilities asked for in the two situations? What methods did you use? b. What did you feel when you were listing the answers? Were you able to list all the possibilities asked for? How did you ensure that your list was complete? What method(s) did you use in order to give the accurate count? Why do you think there is a need to know the number of possible ways a certain task can be done? You will find this out as you go through this lesson. Answer the following questions. 1. Ten runners join a race. In how many possible ways can they be arranged as first, second, and third placers? 2. If Jun has 12 T-shirts, 6 pairs of pants, and 3 pairs of shoes, how many possibilities can he dress himself up for the day? 3. In how many ways can Aling Rosa arrange 6 potted plants in a row? 4. How many four-digit numbers can be formed from the numbers 1, 3, 4, 6, 8, and 9 if repetition of digits is not allowed? Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 16. D EPED C O PY 285 5. If there are 3 roads from Town A to Town B and 4 roads from Town B to Town C, in how many ways can one go from Town A to Town C and back to Town A, through Town B, without passing through the same road twice? 6. Suppose that in a certain association, there are 12 elected members of the Board of Directors. In how many ways can a president, a vice president, a secretary, and a treasurer be selected from the board? 7. In how many ways can you place 9 different books on a shelf if there is space enough for only 5 books? 8. You want to order your lunch from the school canteen, which offers student meals consisting of 1 cup of rice, 1 meat dish, and 1 vegetable dish. How many choices do you have for your meal if there are 3 choices of meat dishes and 2 choices of vegetable dishes? 9. In how many ways can 5 people arrange themselves in a row for picture taking? 10. A dress-shop owner has 8 new dresses that she wants to display in the window. If the display window has 5 mannequins, in how many ways can she dress them up? Questions: a. How did you find the answer to each of the questions? What previously learned principle did you apply? b. Show and explain how you answered each item. In the activity you have done, were you able determine the exact number of ways of doing each task or activity described? What mathematics concept or principle did you use? How was that principle applied? Some of these tasks or activities share similarities or differ from others in some sense. How do they differ? You will find out as you go through the next sections of this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 17. D EPED C O PY 286 Take a second look at the 10 situations in Activity 2. Determine if in doing the activity or task, arrangement or order of selecting the objects is important; that is, whether a different order or arrangement means a different result. Write your answers on a manila paper and be ready to share them to the class. Questions: 1. In which situations in Activity 2 is order or arrangement of the selection important? 2. Justify your answer by giving an example for each situation. 3. In performing a certain task where order or arrangement is important, what do you call each possible arrangement? Perform the following activity using four number cards with different digits. Follow all instructions and write all your answers on a clean sheet of paper. Then, complete the table and answer the questions that follow. A. Get any two number cards (Example: 1 and 2). 1. a. Arrange the cards using 1 piece at a time. Example: 1 2 ways 2 b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made. 2. a. Arrange the cards using both pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made. B. Get any three number cards. 1. a. Arrange the cards using 1 piece at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using 1 card at a time from the 3 given cards. Activity 4: Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 18. D EPED C O PY 287 2. a. Arrange the cards using 2 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using 2 cards at a time from the 3 given cards. 3. a. Arrange the cards using all 3 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made, using all the 3 cards at a time. C. Get the four number cards. 1. Repeat Steps B.1 to 3. 2. a. Arrange the objects using all 4 pieces at a time. b. Illustrate or describe each arrangement. c. Count the number of arrangements you have made using all the 4 number cards at a time. Tabulate all results. Results: Number of Objects (n) Number of Objects Taken at a Time (r) Number of Possible Arrangements 2 1 2 2 2 3 1 3 2 3 3 4 1 4 2 4 3 4 4 Questions: a. What do you call each arrangement? b. Can you find any pattern in the results? c. Can you think of other ways of finding these answers? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 19. D EPED C O PY 288 How did you find the preceding activities? Are you ready to learn about permutations? From the activities you have done, you recalled the Fundamental Counting Principle which is an essential tool in under- standing about arrangement, or permutations. You also identified tasks in which the arrangement or order is important. How can permutations help in solving real-life problems or in making conclusions and decisions? You will find out in the next activities. Before doing these activities, read and understand first some important notes on permutations including the examples presented. How do we find the permutations of objects? Suppose we have 6 different potted plants and we wish to arrange 4 of them in a row. In how many ways can this be done? We can determine the number of ways these plants can be arranged in a row if we arrange only 4 of them at a time. Each possible arrangement is called a permutation. The permutation of 6 potted plants taken 4 at a time is denoted by P(6, 4), 6P4 , P6,4 , or 6 4 P Similarly, if there are n objects which will be arranged r at a time, it will be denoted by P(n, r) . The permutation of n objects taken r at a time is denoted by P(n, r). In some books, it is also denoted by nPr, Pn,r , or n r P . In this learning material, we will use the first notation. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 20. D EPED C O PY 289 How do we find the number of permutations of n objects taken r at a time? Study the following illustrations: Example 1. Given the 4-letter word READ. In how many ways can we arrange its letters, 3 at a time? Solution: There are 3 positions to be filled. We write the horizontal marks: _ _ _. On each horizontal mark, we write the number of choices we have in filling up the said position. There are 4 choices for the first position. After the first position is filled, there are 3 choices left for the second position. After the first and second positions are filled, there are 2 choices left for the third position. So the three entries on the horizontal marks would be 4 3 2 We apply the Fundamental Counting Principle and multiply: 4 ⋅ 3 ⋅ 2 = 24. Thus, there are 24 possible ways of arranging the 4 letters of READ taking 3 of them at a time. We also say the 4 letters of READ, taken 3 at a time, have 24 permutations. One of them is R-E-D. Verify this result. Let us take a second look at the example above. Remember: n = 4, r = 3. 4⋅ 3 ⋅ 2 = n(n – 1)(n – 2) Notice that the first factor is n, and the succeeding factors decrease by 1 each time. Look at the last factor: n - 2 = n – (3 – 1) = n – (r – 1) or n – r + 1. Take note also that there are r factors in all, starting with n. Example 2. In a school club, there are 5 possible choices for the president, a secretary, a treasurer, and an auditor. Assuming that each of them is qualified for any of these positions, in how many ways can the 4 officers be elected? Solution: P(5, 4) = 5 ⋅ 4 ⋅ 3 ⋅ 2 = 120 ways The number of permutations of n objects taken r at a time, P(n, r), where n ≥ r is: P(n, r) = n(n – 1)(n – 2) ⋅⋅⋅ (n – r + 1) All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 21. D EPED C O PY 290 Note that there are r factors. From what principle was the formula derived? Can you now answer the problem earlier? Suppose we have 6 potted plants and we wish to arrange 4 of them in a row. In how many ways can this be done? What if there are 5 objects to be arranged and we would arrange all of them every time? That is, n = 5, and r = 5. Example 3. In how many ways can 5 people arrange themselves in a row for picture taking? Solution: n = 5, r = 5 P(5, 5) = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 (5 factors) = 120 possible pictures In the third example, we used all the numbers from n = 5 down to 1. Another way of writing 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 is 5! (read as 5 factorial) Similarly, 4! = 4⋅ 3⋅ 2⋅ 1 = 24 8! = 8 ⋅7⋅ 6⋅ 5⋅ 4 ⋅3 ⋅2 ⋅1 = 40 320 3! = 3⋅ 2⋅ 1 = 6 10! = 10⋅ 9 ⋅8⋅ 7⋅ 6 ⋅5⋅ 4 ⋅3⋅ 2⋅ 1= 3 628 800 1! = 1 For convenience, we define 0! = 1. Thus, in example 3, P(5, 5) = 5! = 120. Also, P(8, 8) = 8! = 40 320 P(4, 4) = 4! = 24. In addition, since P(n, r) = n(n - 1)(n - 2)⋅⋅⋅(n - r +1) = )!( )!1)(2)...(1)(( rn rnrnnnn   Then, P(n, r) = )!( ! rn n  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 22. D EPED C O PY 291 Remember: The permutation of n objects taken r at a time is: P(n, r) = )!( ! rn n  , n ≥ r and the permutation of n objects taken all at a time is: P(n, n) = n! Notice that, in the previous examples, the objects to be arranged are all distinct. Suppose some of the objects to be arranged are not distinct, that is, some are identical. Study the following examples. Example 4. Find the number of permutations of the letters of the word EVEN. Solution: There are 4 letters of the word. Initially, suppose these letters are distinct, then the number of permutation is P(4, 4) = 4!. However, we have to take into consideration that the 2 E’s are alike. We cannot distinguish between the 2 E’s. How can we arrange the 4 letters differently if the 2 E’s could be differentiated from one another? Instead of using subscripts, let us make one E in upper case and the other in lower case (e). We will have: EVeN eVEN ENVe ENeV VNEe NEVe EVNe eVNE ENeV eNEV NEVe NeVE EeVN eEVN VEeN VeEN NEeV NeEV EeNV eENV VENe VeNE NVEe NVeE How many arrangements are there? Can you think of other possible arrangements? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 23. D EPED C O PY 292 Now, these two E’s are the same, and so the two entries in each box are also the same. Take note that for each permutation like E-V-E-N, there are 2, or 2! ways of arranging the two E’s without changing the arrangement of the others. The duplications are eliminated by dividing 4! or 24 by the number of ways of arranging the 2 E’s which is 2! So, out of the 24 arrangements, the number of distinct or distinguishable permutations, P, would be: P = (4,4) 2! P = 4! 2 = 24 2 = 12 distinguishable permutations Example 5. Find the number of permutations of the letters of the word STATISTICS. Solution: There are 10 letters of the word. Assuming that the letters are distinct, there are P(10, 10) = 10! permutations. However, we have to take into consideration that the 3 S’s are alike, the 3 T’s are alike, and the 2 Is are also alike. The permutations of the 3 S’s is P(3, 3) = 3!. The permutations of the 3 T’s is P(3, 3) = 3! The permutation of the 2 T’s is P(2, 2) = 2! So we must divide 10! by 3! 3! 2! in order to eliminate the duplicates. Thus, P = 10! 3! 3! 2! = 50 400 permutations This leads to the next rule. The number of distinguishable permutations, P, of n objects where p objects are alike, q objects are alike, r objects are alike, and so on, is P = ! ! ! !... n p q r All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 24. D EPED C O PY 293 Let us now consider arrangement of objects in a circle, which we call circular permutations. Example 6. a. In how many ways can 3 people be seated around a circular table? Solution: n = 3 (a) (b) Notice that the arrangement 1-2-3 in (a) above is the same as 2-3-1 and 3-1-2; meaning these 3 permutations are just one and the same. Notice also that the arrangements 1-3-2, 3-2-1, and 2-1- 3 are also just the same as seen in (b). So the circular permutations, P, of 3 objects is: P  6 3  3! 3   3 2! 3  2! 1 2 3 1 3 2 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 25. D EPED C O PY 294 We can also look at it this way: If there are 3 seats around the circular table, we can assign one of the persons to a fixed seat and determine the number of ways that the other two can be arranged. This way, we avoid counting again an arrangement that resulted from a mere rotation. Thus, the number of circular permutations of 3 objects is (3 – 1)! or 2! and the circular permutations of n objects is (n – 1)!. b. In how many ways can 4 people be seated around a circular table? Solution: (n = 4) Let us call the 4 people A, B, C, D The arrangements are: A-B-C-D A-B-D-C A-C-B-D A-C-D-B A-D-B-C A-D-C-B B-C-D-A B-D-C-A C-B-D-A C-D-B-A D-B-C-A D-C-B-A C-D-A-B D-C-A-B B-D-A-C D-B-A-C B-C-A-D C-B-A-D D-A-B-C C-A-B-D D-A-C-B B-A-C-D C-A-D-B B-A-D-C Observe that all the arrangements falling on the same column are just the same because the 4 people are supposed to be seated around a circular table. There are 24 arrangements in the list. Again, the circular permutations, P, of 4 objects is: P = 24 4 based on the list made = 4! 4 = 4 !34  = 3! Or simply, P = (n - 1)! = (4 - 1)! = 3! = 6 The permutation of n objects arranged in a circle is P = (n - 1)! All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 26. D EPED C O PY 295 Know more about permutations through these websites. http://mathworld.wolfram.com/circularpermutations.ht ml http://www.mathsisfun.com/combinatorics/combinatio ns-permutations.html http://users.math.yale.edu/~anechka/math107/lecture 13.pdf Your goal in this section is to apply the key concepts of permutations. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided. Solve for the unknown in each item, and then answer the questions that follow. 1. P(6, 6) = ___ 6. P(8, r) = 6 720 2. P(7, r) = 840 7. P(8, 3) = ___ 3. P(n, 3) = 60 8. P(n, 4) = 3024 4. P(n, 3) = 504 9. P(12, r) = 1320 5. P(10, 5) = ___ 10. P(13, r) = 156 Questions: a. How did you calculate the different permutations? b. What mathematics concepts or principles did you apply to solve each permutation? c. Did you find any difficulty in finding the answers? What technique or strategy can you think of to facilitate your way of solving? How did you find the preceding activity? Was it easy? I am sure it was! Find out from your peers if you had the same answers or strategies in solving. If not, discover why. In the next activity, you are going to apply the concept of permutations to solve real-life problems. Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 27. D EPED C O PY 296 Answer each permutation problem completely. 1. A teacher wants to assign 4 different tasks to her 4 students. In how many possible ways can she do it? 2. In a certain general assembly, three major prizes are at stake. In how many ways can the first, second, and third prizes be drawn from a box containing 120 names? 3. In how many different ways can 5 bicycles be parked if there are 7 available parking spaces? 4. How many distinguishable permutations are possible with all the letters of the word ELLIPSES? 5. There are 8 basketball teams competing for the top 4 standings in order to move up to the semi-finals. Find the number of possible rankings of the four top teams. 6. In how many different ways can 12 people occupy the 12 seats in a front row of a mini-theater? 7. Find the number of different ways that a family of 6 can be seated around a circular table with 6 chairs. 8. How many 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition is allowed? 9. If there are 10 people and only 6 chairs are available, in how many ways can they be seated? 10. Find the number of distinguishable permutations of the digits of the number 348 838. Activity 6: RAFFLE DRAW All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 28. D EPED C O PY 297 Answer each permutation problem completely. 1. In how many different ways can you arrange 8 figurines on a shelf? 2. There are 4 different Mathematics books and 5 different Science books. In how many ways can the books be arranged on a shelf if a. there are no restrictions? b. books of the same subject must be placed together? c. if they must be placed alternately? 3. Five couples want to have their pictures taken. In how many ways can they arrange themselves in a row if a. couples must stay together? b. they may stand anywhere? 4. There are 12 people in a dinner gathering. In how many ways can the host (one of the 12) arrange his guests around a dining table if a. they can sit on any of the chairs? b. 3 people insist on sitting beside each other? c. 2 people refuse to sit beside each other? 5. A teacher drew a number line on the board and named some points on it. The teacher then asked the class to list all the rays in the figure. A student answered 30 rays in all and the teacher said it was correct. How many points were named in the figure? How were the activities done so far? Were you able to answer all of the exercises? I hope you just did successfully! Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 29. D EPED C O PY 298 In this section, permutation was discussed, including permutation with identical objects, circular permutation, and its real-life applications. You may go back to the previous sections and compare your initial ideas about the concept of permutation. Were these further clarified? Now that you have learned the basic ideas about permutations, let us deepen your understanding and study the next section. In this section, you are going to think deeper and test further your understanding of permutations. After doing the following activities, you should be able to answer the important question: How are permutations used in forming conclusions and in making wise decisions? Answer the following questions completely. 1. How do you determine if a situation or problem involves permutations? 2. Differentiate between permutation in general (n objects taken r at a time), circular permutation, and distinguishable permutation (when some objects are alike). 3. a. Find the number of permutations of n objects when arranged (n - 1) at a time for any positive integer n. b. Find the number of permutations of n objects when taken all at a time. c. Compare the two (a and b). Explain why the answers still make sense. 4. Going back to the “combination” lock problem on page 285, suppose that the lock contained the 4 digits 1, 4, 7, 9 but you totally forgot the order in which they come. What are you going to do? How can you apply your knowledge of permutations here? Elaborate. Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 30. D EPED C O PY 299 In this section, the discussion was about your understanding of permutations and how they are illustrated and used in real life. What new realizations do you have about permutations? How would you connect these realizations to real life? Write an entry in your journal describing how much you already learned about permutations and their applications. Include also whatever points in the lesson in which you still need clarifications, and work on these with your teacher. Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which will demonstrate your understanding of permutations. Do the following tasks by group. A. Give 3 examples of problems or situations in real life that involve permutations. In each example, 1. explain the problem or situation. 2. solve the problem. 3. discuss how you can use these sample situations in your daily life, especially in formulating conclusions and/or making decisions. B. It is in international summits that major world decisions happen. Suppose that you were the overall in charge of the seating in an international convention wherein 12 country-representatives were invited. They are the prime ministers/presidents of the countries of Canada, China, France, Germany, India, Japan, Libya, Malaysia, Philippines, South Korea, USA, and United Kingdom. 1. If the seating arrangement is to be circular, how many seating arrangements are possible? 2. Create your own seat plan for these 12 leaders based on your knowledge of their backgrounds. Discuss why you arranged them that way. Activity 10: Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 31. D EPED C O PY 300 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with two or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately , shows in- depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately , shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 In this section your task was to give examples of real-life situations where permutation is illustrated. You also formulated and solved problems related to permutation. SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about permutations and its applications in real-life situations. The lesson provided you with opportunities to identify situations that describe permutations and differentiate them from those that do not. You were also given a chance to perform practical activities for you to further understand the topic. In addition, you were given the opportunity to formulate and solve problems on permutations and apply the knowledge to formulating conclusions and making decisions. Your understanding of this lesson as well as the other Mathematics concepts previously learned will help you learn the next topic, combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 32. D EPED C O PY 301 Start Lesson 2 of this module by assessing your knowledge of the Fundamental Counting Principle and permutations. This knowledge and skill will help you understand combinations, which will further help you in forming conclusions and in making decisions. To be able to do this, perform each activity that follows. Seek the assistance of your teacher and peers if you encounter any difficulty. Have your work checked by your teacher. Solve each problem below and answer the questions that follow. 1. If your school cafeteria offers pork, beef, chicken, and fish for main dish, chop suey, pinakbet, and black beans for vegetable dishes, banana and pineapple for dessert, and tea, juice, and softdrinks for beverage, in how many ways can you choose your meal consisting of 1 cup of rice, 1 main dish, 1 vegetable dish, 1 beverage, and 1 dessert? 2. You were tasked to take charge of the auditions for the female parts of a stage play. In how many possible ways can you form your cast of 5 female members if there were 15 hopefuls? 3. If ice cream is served in a cone, in how many ways can Abby choose her three-flavor ice cream scoop if there are 6 available flavors? 4. If each Automated Teller Machine card of a certain bank has to have 4 different digits in its passcode, how many different possible passcodes can there be? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 33. D EPED C O PY 302 5. How many possible permutations are there in the letters of the word PHILIPPINES? 6. In how many ways can a group of 6 people be seated around a table if 2 of them insist on sitting beside each other? 7. In how many ways can 4 students be seated in a classroom if there are 7 available seats? 8. In how many ways can 5 English books and 4 Mathematics books be placed on a shelf if books of the same subject are to be together? 9. A family catering business prides itself with its offerings of delicious meals and other food requirements. If you were one of the staff or key persons in this business, how can you apply your knowledge of permutation and combination to further improve your business? 10. Due to the huge population of Mapayapa High School, one of the problems encountered is the big crowd through the gates during the early morning and at dismissal time in the afternoon. If you were one of the administrators of the school, what step can you suggest to remedy this problem? a. How did you find the number of ways asked for in each item? What mathematics concepts or principles did you apply? How did you apply these concepts or principles? b. Which situations above illustrate permutations? Which do not? Why? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 34. D EPED C O PY 303 Were you able to find what was required in each problem? Were you able to recall and apply the different mathematics concepts or principles in solving each problem? Why do you think it is important to be able to perform such mathematical tasks? You will find this out as you go through this lesson. Study the tasks or activities below, and then answer the questions that follow. 1. Choosing 5 questions to answer out of 10 questions in a test 2. Opening a combination lock 3. Winning in a contest 4. Selecting 7 people to form a Student Affairs Committee 5. Forming triangles from 6 distinct points in which no 3 points are collinear 6. Assigning seats to guests at dinner 7. Drawing a set of 6 numbers in a lottery containing numbers 1 to 45 8. Entering the PIN (Personal Identification Number) of your ATM card 9. Selecting 3 posters to hang out of 6 different posters 10. Listing the elements of subsets of a given set Questions: a. In which tasks/activities above is order or arrangement important? Give an example to illustrate each answer. b. In which tasks/activities is order not important? Give an example to illustrate each answer. In the activity you have just done, were you able to identify situations that involve permutations and those that do not? The latter are called combinations and you will learn more about them in the next activities. Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 35. D EPED C O PY 304 Perform the following activity using any 5 different fruits (mango, guava, banana, pomelo, avocado, or any fruits available in your place). Follow all instructions and write all your answers on a clean sheet of paper. Then, complete the table and answer the questions that follow. A. Get 2 fruits (ex. mango and banana). 1. a. Select a fruit 1 piece at a time. Do all possible selections. b. Illustrate or describe each selection you made. Example: mango banana  2 ways c. Count the number of different selections you have made. 2. a. Select the fruits 2 pieces at a time. b. Illustrate or describe each selection. c. Count the number of different selections you have made. B. Get 3 fruits. 1. a. Select a fruit 1 piece at a time. Do all possible selections. b. Illustrate or describe each selection. c. Count the number of different selections you have made when using 1 object at a time from the 3 given fruits. 2. a. Select 2 fruits at a time. b. Illustrate or describe each selection. c. Count the number of selections you have made, using 2 objects at a time from the 3 given fruits. 3. a. Select 3 fruits at a time. b. Illustrate or describe each selection. c. Count the number of selections you have made, using all the 3 fruits at a time. C. Get 4 fruits. 1. Repeat Steps B.1 to 3. 2. a. Continue the process until you select 4 fruits at a time. b. Illustrate or describe each selection. c. Count the number of different selections you have made using all the 4 fruits at a time. D. Repeat the same procedure for 5 fruits. Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 36. D EPED C O PY 305 Results: Number of Objects (n) Number of Objects Taken at a Time (r) Number of Possible Selections 2 1 2 2 3 1 3 2 3 3 4 1 4 2 4 3 4 4 5 1 5 2 5 3 5 4 5 5 Questions: 1. Did it matter in what order you selected the objects? 2. Give an example to justify your answer in (1). 3. What do you call each unique selection? 4. Can you find any pattern in the results? 5. Can you think of other ways of finding these answers? How did you find the preceding activities? Are you ready to learn about combinations? From the activities you have done, you identified tasks in which the arrangement or order is important (permutations) and those in which order is not (combinations). You recalled the Fundamental Counting Principle as well as permutation which are essential concepts in understanding combinations. In the next sections of this learning module, keep in mind the important question: How can the concept of combinations help in solving real-life problems or in formulating conclusions and making wise decisions? You will find out when you do the next activities. Before doing these activities, read and understand first some important notes on combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 37. D EPED C O PY 306 Suppose you were assigned by your teacher to be the leader of your group for your project. You were given the freedom to choose 4 of your classmates to be your group mates. If you choose Aira, Belle, Charlie, and Dave, does it make any difference if you choose instead Charlie, Aira, Dave, and Belle? Of course not, because the list refers to the same people. Each selection that you could possibly make is called a combination. On the other hand, if you choose Aira, Belle, Dave, and Ellen, now that is another combination, and it is different from the first combination cited. Remember: If there is a set S with n elements, and if r is a nonnegative integer less than or equal to n, then each subset of S containing r distinct elements is called a combination of S. The number of combinations of n objects taken r at a time is denoted by C(n, r) , nCr , n rC or       r n . In this learning material, we will use the first notation. How do we find the number of combinations of n objects taken r at a time? Suppose now, that you are asked to form different triangles out of 4 points plotted, say, A, B, C, and D, of which no three are collinear. We can see that ABC is the same as BCA and CBA. In the same manner, BCD is the same as CBD and DBC. This is another illustration of combination. The different triangles that can be formed are ABC, ABD, BCD, and CDA. Thus, there are 4 combinations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 38. D EPED C O PY 307 How can we find the number of combinations more systematically? Consider this: If order of the letters is important, then we have the following possibilities: ABC ABD BCD CDA ACB ADB BDC CAD BCA BDA CDB DAC BAC BAD CBD DCA CAB DBA DBC ADC CBA DAB DCB ACD The number of different orders of 4 vertices taken 3 at a time is given by P(4, 3) = 4! (4 3)! = 24. There are 24 possibilities. Since you learned in Geometry that we can name a triangle using its three vertices in any order, then if we look more closely, we can see that all the triangles in the same column are identical. Thus, the actual number of combinations is C(4, 3) = 24 6 or ( )4, 6 3P or ( 3 4, ! 3)P . (Equation 1) Notice that 6 or 3! is the number of ways of arranging 3 objects taken all at a time. We divided by 3! to eliminate duplicates. Note : There are 4 objects (A,B,C,D)  n = 4 They are selected 3 at a time  r = 3. And so equation (1) becomes C(n,r) = n,P( r) r! . Since P(n, r) = )!( ! rn n  , then C(n, r) = n,P( r) r! = ! )!( ! r rn n  = )!(! ! rnr n  . All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 39. D EPED C O PY 308 Remember: The combination of n objects taken r at a time is: C(n, r) = ! ! ! n r (n r ) , n ≥ r ≥ 0 Example 1. In how many ways can a committee consisting of 4 members be formed from 8 people? Solution 1: (Using the formula) n = 8, r = 4 C(n,r) = ! ! ( )! n r n r = 8! 4! (8 4)! = 2 8 7 6 5 4 3 2 1 4 3 2 1 4 3 2 1               = 7 2  5 after common factors are cancelled = 70 ways Solution 2: Based on the discussion on the previous page, C(n,r) = ( , ) ! P n r r =   8 7 6 5 4! =       2 8 7 6 5 4 3 2 1 = 7  2 5 after common factors are cancelled = 70 ways There are r = 4 factors Permutation of n objects taken r at a time Permutation of r objects All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 40. D EPED C O PY 309 Example 2. How many polygons can be possibly formed from 6 distinct points on a plane, no three of which are collinear? Solution: The polygon may have 3, 4, 5, or 6 vertices. Thus, the number N of possible polygons is: N = C(6, 3) + C(6, 4) + C(6, 5) + C(6, 6) = 6 5 4 3!   + 6 5 4 3 4!    + 6 5 4 3 2 5!     + 6 5 4 3 2 1 6!      = 6 5 4 3 2 1     + 6 5 4 3 4 3 2 1       + 6 5 4 3 2 5 4 3 2 1         + 6 5 4 3 2 1 6 5 4 3 2 1           = 5 ∙ 4 + 5 ∙ 3 + 6 + 1 = 20 + 15 + 7 N = 42 possible polygons. Verify this answer by using the formula for combinations. Read more about combinations in these websites. www.beatthegmat.com/mba/2013/09/27/does-order- matter-combinations-and-non-combinations http://www.mathsisfun.com/combinatorics/combinatio ns-permutations.html http://users.math.yale.edu/~anechka/math107/lecture 13.pdf http://voices.yahoo.com/the-importance- permutations-combinations-in-10262.html All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 41. D EPED C O PY 310 Your goal in this section is to apply the key concepts of combinations. Use the mathematical ideas and the examples presented in the preceding section to answer the given activities. Study the following situations. Then, answer the questions that follow. 1. Determining the top three winners in a Science Quiz Bee 2. Forming lines from six given points with no three of which are collinear 3. Forming triangles from 7 given points with no three of which are collinear 4. Four people posing for pictures 5. Assembling a jigsaw puzzle 6. Choosing 2 household chores to do before dinner 7. Selecting 5 basketball players out of 10 team members for the different positions 8. Choosing three of your classmates to attend your party 9. Picking 6 balls from a basket of 12 balls 10. Forming a committee of 5 members from 20 people Questions: a. In the items above, identify which situations illustrate permutation and which illustrate combination. b. How did you differentiate the situations that involve permutation from those that involve combination? Were you able to differentiate the tasks/situations that involve permutation from those that involve combination? The next activity will provide you with more exercises to improve your calculating skills. Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 42. D EPED C O PY 311 Find the unknown in each item. 1. C(8, 3) = ___ 6. C(10, r) = 120 2. C(n, 4) = 15 7. C(n, 2) = 78 3. C(8, r) = 28 8. C(11, r) = 165 4. C(9, 9) = ___ 9. C(8, 6) = ___ 5. C(n, 3) = 35 10. C(14,10) = ___ How did you find the activity? Was it easy? I guess it was! Go on to the next activity to apply your knowledge of combinations in real-life situations. Solve the following problems completely. (Choose a partner.) 1. If there are 12 teams in a basketball tournament and each team must play every other team in the eliminations, how many elimination games will there be? 2. If there are 7 distinct points on a plane with no three of which are collinear, how many different polygons can be possibly formed? 3. How many different sets of 5 cards each can be formed from a standard deck of 52 cards? 4. In a 10-item Mathematics problem-solving test, how many ways can you select 5 problems to solve? 5. In problem number 4, how many ways can you select the 5 questions if you are required to answer question number 10? 6. In how many ways can a committee of 5 be formed from 5 juniors and 7 seniors if the committee must have 3 seniors? 7. From a population of 50 households, in how many ways can a researcher select a sample with a size of 10? 8. A box contains 5 red balls, 7 green balls, and 6 yellow balls. In how many ways can 6 balls be chosen if there should be 2 balls of each color? Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 43. D EPED C O PY 312 9. From 7 Biology books and 6 Chemistry books, in how many ways can one select 2 Biology and 2 Chemistry books to buy if all the said books are equally necessary? 10. Mrs. Rivera’s business is gown rental and sale. She decided one day that she would display her 10 newest gowns in her shop’s window to attract customers. If she only had 5 mannequins and planned to change the set of gowns every 2 days, how many days will have to pass before she runs out of a new set to display? Were you able to do the activity successfully? I am sure you did! Go on to the next activity to see if you can solve the following set of permutation and combination problems. Solve the following permutation and combination problems. 1. In how many ways can you arrange 5 Mathematics books, 4 Science books, and 3 English books on a shelf such that books of the same subject are kept together? 2. In how many ways can 6 students be seated in a row of 6 seats if 2 of the students insist on sitting beside each other? 3. In a gathering, the host makes sure that each guest shakes hands with everyone else. If there are 25 guests, how many handshakes will be done? Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 44. D EPED C O PY 313 4. Suppose you are the owner of a sari-sari store and you want to put 12 pieces of canned goods in a row on the shelf. If there are 3 identical cans of meat loaf, 4 identical cans of tomato sauce, 2 identical cans of sardines, and 3 identical cans of corned beef, in how many different ways can you display these goods? 5. A soloist is auditioning for a musical play. If she is required to sing any three of the 7 prepared songs, in how many ways can she make her choice? 6. In a dance contest, each participating group must perform 3 kinds of dance. If there are 4 choices for ballroom dance, 8 choices for foreign dance, and 5 choices for hip-hop, in how many possible ways can a dance group select their piece? 7. If 3 marbles are picked randomly from a jar containing 6 red marbles and 8 green marbles, in how many possible ways can it happen that at least 2 of the marbles picked are green? 8. There are 10 identified points on a number line. How many possible rays can be drawn using the given points? 9. You are transferring to a new house. You have a collection of books but you cannot take them all with you. In how many ways can you select 7 books out of 10, and then arrange these books on a shelf if there is space enough for only 5 books? 10. At Enzo’s Pizza Parlor, there are seven different toppings, where a customer can order any number of these toppings. If you dine at the said pizza parlor, with how many possible toppings can you actually order your pizza? How did you find all the activities you have done? I hope you have answered the exercises correctly. In this section, combination was discussed, including its real-life applications. You were also given the opportunity to differentiate combination from permutation. Now that you have learned the basic ideas about combination, let us deepen your understanding and study the next section. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 45. D EPED C O PY 314 In this section you are going to think more deeply and test further your understanding of combinations. After doing the following activities, you should be able to answer the important question: How are combinations used in forming conclusions and making wise decisions? Answer the following questions completely. 1. How do you determine if a situation involves combinations? 2. To find the total number of polygons that can be formed from 7 points on a plane with no three of which are collinear, Joy answered: C(7, 3) = !3!4 !7 = !3!4 !4567  = 123 567   = 35 different polygons Is Joy correct? Justify your answer. 3. (a) In how many ways can the 12 members of the Board of Directors (BOD) be chosen from 12 parent-nominees and 7 teacher-nominees if there must be 8 parents in the BOD? (b) After the 12 members are chosen, in how many ways can they elect among themselves the 7 top positions (president, vice president, and others)? 4. DAMATH is a board game that incorporates mathematical skills in the Filipino game Dama. In a school DAMATH tournament, there are 28 participants who are divided into 7 groups. Each participant plays against each member of his group in the eliminations. The winner in each group advances to the semi-finals where they again compete with each other. The five players with the most number of wins proceed to the final round and play against each other. Assume that there are no ties. a. What is the total number of games to be played in the eliminations? b. How many matches will be played in the final round? Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 46. D EPED C O PY 315 c. In how many possible ways can the top five players in the semi- finals come up? d. In how many possible ways can the 1st, 2nd, and 3rd placer be declared in the final round? e. How many matches will be played altogether? In this section, the discussion was about your understanding of combinations and how they are illustrated and used in real life. What new learnings do you have about combinations? How can these learnings be applied in real life? Write an entry in your journal describing how much you have learned about combinations, and how these can be applied to real life. Add also the parts of the lesson that you still find confusing, if any. Work on these difficulties with your teacher. Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which requires that you demonstrate your understanding of combinations. Answer the following problems completely. Give 3 examples of situations in real life that illustrate combinations. In each situation, 1. formulate a problem. 2. solve the problem. 3. explain how this particular problem may help you in formulating conclusions and/or making decisions. Activity 10: Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 47. D EPED C O PY 316 Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes. 4 Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately , shows in-depth comprehension of the pertinent concepts and/or processes 3 Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately , shows comprehension of major concepts although neglects or misinterprets less significant ideas or details 2 Poses a problem and finishes some significant parts of the solution and communicates ideas accurately but shows gaps on theoretical comprehension 1 Poses a problem but demonstrates little comprehension, not being able to develop an approach Source: D.O. #73, s. 2012 SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about combinations and their applications in real life. Through the lesson, you were able to identify situations that describe combinations and differentiate them from those that do not. You were also given the opportunity to perform practical activities to further understand the topic, formulate related real-life problems, and solve these problems. You also applied your knowledge to formulating conclusions and making wise decisions. Your understanding of this lesson, combined with other previously learned Mathematics concepts will help you understand the next lesson, probability. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 48. D EPED C O PY 317 GLOSSARY OF TERMS Circular permutation – the different possible arrangements of objects in a circle. The number of permutations, P, of n objects around a circle is given by P = (n – 1)!. Combinations – the number of ways of selecting from a set when the order is not important. The number of combinations of n objects taken r at a time is given by C(n, r) = !)!( ! rrn n  , n ≥ r. Distinguishable permutations – refers to the permutations of a set of objects where some of them are alike. The number of distinguishable permutations of n objects when p are alike, q are alike, r are alike, and so on, is given by P = !...!! ! rqp n . Fundamental Counting Principle – states that if activity A can be done in n1 ways, activity B can be done in n2 ways, activity C in n3 ways, and so on, then activities A, B, and C can be done simultaneously in  321 nnn ways. Permutations – refers to the different possible arrangements of a set of objects. The number of permutations of n objects taken r at a time is: P(n, r) = )!( ! rn n  , n ≥ r. n-Factorial – the product of the positive integer n and all the positive integers less than n. n! = n(n – 1)(n – 2) … (3)(2)(1). REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bennett, J. & Chard, D., et al. (2005). Pre-Algebra. Texas: Holt, Rinehart and Winston. Bhowal, M. & Barua, P. (2008). Statistics: 2nd ed. New Delhi: Kamal Jagasia. Leithold, L. (2002). College Algebra and Trigonometry. Singapore: Pearson Education Asia Pte. Ltd. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 49. D EPED C O PY 318 McCune, S. (2010). Statistics. New York: The Mc-Graw-Hill Companies, Inc. Kelly, W. and Donnelly, R. (2009). The Humungous Book of Statistics Problems. New York: Penguin Group (USA), Inc. Spiegel, M.R. & Stephens, L.J. (2008). Schaum’s Outline of Theory and Problems of Statistics 4th ed. New York: The Mc-Graw-Hill Companies, Inc. Website Links as References and Sources of Learning Activities: BBC News About International Summits. Jonathan Powell. May 29, 2012. www.bbc.co.uk/news/magazine-18237721 Accessed on Feb. 27, 2014 Circular Permutation. Weisstein, Eric W. From Mathworld - A Wolfram Web Resource. http://mathworld.wolfram.com/circularpermutations.html Accessed on Feb. 26, 2014 Combinations and Permutations http://www.mathsisfun.com/combinatorics/combinations-permutations.html Accessed on Feb. 26, 2014 Does Order Matter- Combinations and Non-Combinations. Brent Hanneson. Sept. 27, 2013. www. beatthegmat.com/mba/2013/09/27/does-order-matter- combinations-and-non-combinations-partiii. Accessed on Feb. 27, 2014 Mathematics in the Real World. http://users.math.yale.edu/~anechka/math107/lecture13.pdf. Accessed on Feb. 25, 2014 The Fundamental Counting Principle and Permutations. http://www.classzone.com/eservices/home/pdf/student/LA212AAD.pdf. Accessed on Feb. 25, 2014 The Importance of Permutations and Combinations in Modern Society. Valerie Hansen.Nov.7, 2005. http://voices.yahoo.com/the-importance- permutations-combinations-in-10262.html?cat=41 Accessed on Feb. 26, 2014 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 50. D EPED C O PY 319 I. INTRODUCTION It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge. The most important questions of life are indeed, for the most part, really only problems of probability. Pierre Simon de Laplace Maybe, you are familiar with games of chance such as lotto. You have also learned that the likelihood of winning in any game of chance can be estimated using probability. However, do you not know that the theory of probability is not only for games of chance? Probability is very useful in many practical and important fields in science, engineering, economics, and social sciences. Why do you think is the study of probability so important to the advancement of science and technology? You will learn the reasons when you study this module. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 51. D EPED C O PY 320 II. LESSONS AND COVERAGE Lesson 1 – Probability of Compound Events  Union and Intersection of Events  Mutually Exclusive and Not Mutually Exclusive Events Lesson 2 – Probability of Independent Events Lesson 3 – Conditional Probability In these lessons, you will learn to: Lesson 1  illustrate events, and union and intersection of events  illustrate the probability of a union of two events and intersection of events  illustrate and find probabilities of mutually exclusive events Lesson 2  illustrate independent and dependent events  find probabilities of independent and dependent events Lesson 3  identify conditional probabilities  solve problems on conditional probabilities Probability of Compound Events Probability of Compound Events Union and Intersection of Events Probability of Independent Events Conditional Probability Mutually Exclusive and Not Mutually Exclusive Events All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 52. D EPED C O PY 321 III. PRE-ASSESSMENT Part I: Multiple Choice Answer all of the following questions the best that you can. As much as possible, provide your own solution. Take note of the items that you were not able to answer correctly and find out the right answer as you go through this module. 1. Brian likes to wear colored shirts. He has 10 shirts in the closet. Three of these are blue, four are in different shades of red, and the rest are of mixed or different colors. What is the probability that he will wear a blue or a red shirt? A. 7 4 10 10  B. 3 4 10 10  C. 3 7 10 10  D. 7 4 10 10  2. The spinner on the right is spun. What is the probability of a spin that results in an even number or a number less than 4? A. 1 4 B. 3 8 C. 4 8 D. 5 8 3. Jody has four cans of juice – one can of orange, one of pineapple, one of calamansi, and one of guyabano. She chooses three of these cans to take to school. If she chooses calamansi, what is the probability she also chooses pineapple? A. 7 8 B. 3 4 C. 2 3 D. 3 8 4. A man tosses a fair coin eight times, and observes whether the toss yields a head (H) or a tail (T). Which of the following sequences of outcomes yields a head (H) on his next toss? (I) T T T T T T T T (II) H H T H T T H H A. I C. Neither I nor II B. II D. Either I or II All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 53. D EPED C O PY 322 5. A baby has 5 blocks in a box. One block is red, one is yellow, one is green, one is blue, and one is black. The baby pulls out a block, looks at it, and puts it back in the box. If he does this 4 times before he gets bored and crawls away, what is the probability that the 4 blocks selected are all of the same color? A. 4 5 5 B. 4 1 5 C. 4 4 5 D. 4 2 5 6. A box contains 4 red balls and 6 blue balls. A second box contains 16 red balls and an unknown number of blue balls. A single ball is drawn from each box. The probability that both balls are of the same color is 0.44. How many blue balls are there in the second box? A. 4 B. 20 C. 24 D. 44 7. A family has two children. Suppose that the birth of each child is an independent event and that it is equally likely to be a boy or a girl. Let C denote the event that the family has one boy and one girl. Let D denote the event that the family has at most one girl. Which of the following must be true about events C and D? A. C and D are independent events. B. C occurs given that D does not occur. C. C and D are not independent events. D. C and D are mutually exclusive events. 8. A nationwide survey revealed that 42% of the population likes eating pizza. If two people are randomly selected from the population, what is the probability that the first person likes eating pizza while the second one does not? A. 0.42 + (1 − 0.42) C. 1 - 0.42 B. 2 (1 − 0.42) D. 0.42 (1 − 0.42) 9. A married couple agreed to continue bearing a new child until they get two boys, but not more than 4 children. Assuming that each time that a child is born, the probability that it is a boy is 0.5, independent from all other times. Find the probability that the couple has at least two girls. A. 1 2 B. 5 16 C. 5 8 D. 4 15 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 54. D EPED C O PY 323 10.Some street foods were sampled and tested for the presence of disease-causing bacteria or harmful chemicals. A random sample of 200 street foods of various types according to how they are prepared was examined. The table below shows the results: Type of Food Number of Food with Bacteria Only Number of Food with Harmful Chemicals Only Number of Food with Both Bacteria and Harmful Chemicals Total Fried 35 15 18 68 Boiled 46 14 32 92 Grilled 24 8 8 40 Total 105 37 58 200 a.) What is the probability that a street food selected at random is fried? A. 68 200 B. 35 105 C. 50 142 D. 68 142 b.) What is the probability that a food selected at random is both grilled and contained harmful chemical? A. 40 200 B. 58 105 C. 16 142 D. 16 200 c.) What is the probability that a randomly selected food has both bacteria and harmful chemicals? A. 40 200 B. 58 200 C. 16 142 D. 8 58 11.A survey of a group’s sports viewing habits over the last year revealed the following information: i. 28% watched soccer ii. 29% watched basketball iii. 19% watched tennis iv. 14% watched soccer and basketball v. 12% watched basketball and tennis vi. 10% watched soccer and tennis vii. 8% watched all three sports. What percent of the group watched none of the three sports last year? A. 24 B. 36 C. 41 D. 52 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 55. D EPED C O PY 324 12.The probability that a visit to the school clinic is neither due to dental reasons nor medical reasons is 35%. Of those coming to the clinic, 30% are due to medical reasons and 40% are due to dental reasons. What is the probability that a visit to the school clinic is due to both dental and medical reasons? A. 0.05 B. 0.12 C. 0.18 D. 0.25 13.A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and of these 312 men, 102 died from causes related to heart disease. Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. A. 102 625 B. 108 625 C. 312 625 D. 414 625 14.There are four batteries, and one of them is defective. Two are to be selected at random for use on a particular day. Find the probability that the second battery selected is not defective, given that the first was not defective. A. 2 3 B. 1 4 C. 1 3 D. 1 2 For numbers 15 to 16: A sample of 150 plastic pipes were selected and subjected to shock resistance and scratch resistance tests. The results are summarized in the table below. Scratch Resistance Shock Resistance High Low High 125 12 Low 7 6 15.A pipe is selected at random. What is the probability that it has high shock resistance given that it has high scratch resistance? A. 125 132 B. 125 137 C. 137 150 D. 132 150 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 56. D EPED C O PY 325 16.What is the probability that it has high scratch resistance and high shock resistance? A. 125 150 B. 125 137 C. 137 150 D. 132 150 17.An insurance agent offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are 1 4 , 1 3 , and 5 12 , respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage. A. 7 9 B. 1 2 C. 47 144 D. 9 144 18.There are 24 dolphins in an ocean park. The caretaker tags 6 of them with small chips and returns them to the ocean park. The next month, he randomly selects five dolphins from the ocean park. a.) Find the probability that exactly two of the selected dolphins are tagged. A. 2 3 5 6 18 24 C C C B. 3 2 5 6 18 24 C C C C. 3 2 5 6 18 24 P P P D. 2 3 5 6 18 24 P P P b.) What were some of your assumptions in part (a) of this item? Justify your answer. Part II: Problem Solving Read and understand the situations below and solve the problem. Show your complete solution. 19.Varsity Try-Out. Suppose you are applying as a tennis varsity player of a team. To be accepted, you need to play with the team’s good player (G) and top player (T) in three games and win against both G and T in two successive games. You must choose one of the two schedules: playing G, T, G or T, G, T. Which one should you choose? Why? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 57. D EPED C O PY 326 20.Food Preference Survey. A school conducts a survey of their students’ food preference during lunch and gets the following information: i. All students prefer to have at least one viand. ii. 70% of the students prefer to have more than one viand iii. 20% of the students prefer to have fish as viand. iv. Of those students who prefer to have more than one viand, 15% prefer fish as viand. Find the probability that a randomly selected student prefers to have exactly one viand and that is fish. Show your solution. IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of the key concepts of probability of compound events, mutually exclusive events, independent events, and of conditional probability. With these expected knowledge and skills, you should be able to use probability in formulating conclusions and in making decisions. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 58. D EPED C O PY 327 Scoring Rubrics for Problem Solving 1 2 3 4 Problem Solving No strategy is chosen, or a strategy chosen will not lead to a solution. Little or no evidence of en- gagement in the task present A partially correct strategy is chosen, or a correct strategy for solving only part of the task is chosen. Evidence of drawing on some relevant previous knowledge is present, showing some relevant engagement in the task. A correct strategy is chosen based on the mathematical situation in the task. Planning or monitoring of strategy is evident. Evidence of solidifying prior knowledge and applying it in problem- solving is present. Note: At this level a student must achieve a correct answer. An efficient strategy is chosen and progress towards a solution is evaluated. Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered. Evidence of analyzing the situation in mathematical terms and extending prior knowledge is present. Note: At this level a student must achieve a correct answer. Reasoning Arguments are made with mathematical basis. No correct reasoning nor justification for reasoning is present. Arguments are made with some mathematical basis. Some correct reasoning or justification for reasoning is present. Arguments are constructed with adequate math- ematical basis. A systematic approach and/or justification of correct reasoning is present. Deductive arguments are used to justify decisions and may result in formal proofs. Evidence is used to justify and support decisions made and conclusions reached. Communication No awareness of audience or purpose is communicated. No formal mathematical terms or symbolic notations are evi- dent. Some communication of an approach is evident through verbal/written accounts and explanations. An attempt is made to use formal math language. One formal math term or symbolic notation is evident. Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response. Formal math language is used to share and clarify ideas. At least two formal math terms or symbolic notations are evident, in any combination. Communication of argument is supported by mathematical properties. Formal math language and symbolic notation is used to consolidate math thinking and to communicate ideas. At least one of the math terms or symbolic notations is beyond grade level. Connection No connections are made or connections are mathematically or contextually irrelevant. A mathematical connection is attempted but is partially incorrect or lacks contextual relevance. A mathematical connection is made. Proper contexts are identified that link both the mathematics concepts and the situation presented in the task. Mathematical connections are used to extend the solution to other mathematics problems or to gain a deeper understanding of the mathematics concepts. Some examples may include one or more of the following: testing and accepting or rejecting a hypothesis as an explanation of a phenomenon. Representation No attempt is made to construct a mathematical representation. An attempt is made to construct a mathematical representation to re- cord and communicate problem solving but is not accurate. An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions. An appropriate math- ematical representation is constructed to analyze relationships, extend thinking, and clarify or interpret a phenomenon. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 59. D EPED C O PY 328 Begin Lesson 1 of this module by assessing your knowledge and skills of the different mathematics concepts related to counting techniques and probability of simple events as well as concepts of sets you previously studied. These knowledge and skills are important in understanding the probability of compound events. As you go through this lesson, think of this question, Why do you think is the study of probability important in making decisions in real life? Consider the situation below. Use your knowledge on probability in answering the questions that follow. 1. A die is rolled once. Find the probability of obtaining a. a 5. b. a 6. c. an odd number. 2. A box contains 3 red balls, 5 yellow balls, and 2 blue balls. If a ball is picked at random from the box, what is the probability that a ball picked is a a. yellow ball? b. red ball? The above activity helped you recall your knowledge of probability of simple events. If you roll a die, the number that would come up could be 1, 2, 3, 4, 5, or 6. When the die is rolled, it is equally likely to land on one face as on any other. Therefore, the probability of getting a “5” is one out of 6. In symbol, we use P(getting a 5) = 1 6 . Always remember that 1 6 is the probability that any of the faces shows up. Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 60. D EPED C O PY 329 Activities such as rolling a die, tossing a coin, or randomly choosing a ball from a box which could be repeated over and over again and which have well- defined results are called experiments. The results of an experiment are called outcomes. The set of all outcomes in an experiment is called a sample space. An event is a subset of the sample space. Simple Events: Consider rolling a die. a. “Getting a number 5” is called a simple event. b. “Getting a 6” is also a simple event. What about the event of “getting an odd number”? Probability of Simple Events: If each of the outcomes in a sample space is equally likely to occur, then the probability of an event E, denoted as P(E) is given by   number�of�ways�the�event�can�occur number�of�possible�outcomes P E  or   number�of�outcomes�in�the�event number�of�outcomes�in�the�sample�space P E  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 61. D EPED C O PY 330 Consider the situation below. Use the tree diagram given below in answering the questions that follow. A school canteen serves lunch for students. A set of menu consists of 1 type of rice, 1 type of viand, and 1 type of drink. The tree diagram below shows the possible menu combinations. Rice Viand Drink pineapple juice chicken adobo orange juice Fried rice pineapple juice pinakbet orange juice pineapple juice chicken adobo orange juice Steamed rice pinakbet pineapple juice orange juice 1. Give the sample space of combination of rice, viand, and drink. How many possible outcomes are there? 2. List the outcomes of selecting a lunch with pineapple juice. 3. How many outcomes are there for selecting any lunch with pineapple juice? 4. How many outcomes are there for selecting a lunch with steamed rice and with pineapple juice? 5. How many outcomes are there for selecting a lunch with chicken adobo and a pineapple juice? 6. How many outcomes are there for selecting a lunch with pinakbet and an orange juice? A student taking lunch in the canteen is selected at random. 7. What is the probability that the student chose pineapple juice as a drink? 8. What is the probability that the student chose steamed rice and pineapple juice? 9. What is the probability that the student chose chicken adobo and orange juice? 10. What is the probability that the student chose pinakbet and pineapple juice? Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 62. D EPED C O PY 331 Reflect: a. What does the tree diagram tell you? b. How did you determine the sample space? c. Differentiate an outcome from a sample space. Give another example of an outcome. d. Aside from the tree diagram, how else can you find the total number of possible outcomes? e. Describe the outcome in this situation as compared to the events that you studied in grade 8. From the above activity, you were able to recognize that the events in the given situation are not simple events. This is because in finding the sample, you need to find first the sample space using the fundamental counting principle. The events mentioned here are called compound events. Typically, when the sample space is finite, any subset of the sample space is an event. Any subset of the sample space is an event. Since all events are sets, they are usually written as sets (e.g., {1, 2, 3}). Compound Events: Events which consist of more than one outcome are called compound events. A compound event consists of two or more simple events. Example: Finding the probability of “getting a 6 and a 1” when two dice are rolled is an event consisting of (1, 6), (6, 1) as outcomes. The first die falls in 6 different ways and the second die also falls in 6 different ways. Thus, using the fundamental counting principle, the number of outcomes in the sample space is 66 or 36. The outcomes in the sample space are: {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3),…,(6, 5), (6, 6)}. Take note that “getting a 6 and a 1” when two dice are rolled is an event consisting of {(1, 6), (6, 1)} as outcomes. This is a compound event. It is often useful to use a Venn diagram to visualize the probabilities of events. To understand more about the probability of the union and intersection of events, you may proceed to Activity 3. The next activity will help you understand the concept of intersection and union of events. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 63. D EPED C O PY 332 The extracurricular activities in which the senior class at Kananga National High School participate are shown in the Venn diagram below. 1. How many students are in the senior class? 2. How many students participate in athletics? 3. If a student is randomly chosen, what is the probability that the student participates in athletics or drama? 4. If a student is randomly chosen, what is the probability that the student participates only in drama and band? Reflect: a. How were you able to find the total number of students in the senior class? b. How does the concept of set help you in finding the intersection and union of two or more events? c. What are some notations that are used in your study of sets in grade 7 that you can still recall? Do you think these are needed in the study of probability of compound events? Activity 3: Extra-curricular activities participated by senior students All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 64. D EPED C O PY 333 Illustrative Example: The Venn diagram below shows the probabilities of grade 10 students joining either soccer (S) or basketball (B). Use the Venn diagram to find the probabilities. a. P(B) b. P(S) c. P(BS) d. P(BS) e. P(B’S’) To further understand the above problem, read the discussion of the solution. Actually, the diagram does not show or represent the entire sample space for B and S. What is shown are the probabilities. a. To find the P(B), we will add the probability that only B occurs to the probability that B and S occur to get 0.4 + 0.3 = 0.7. So, P(B) = 0.7. b. Similarly, P(S) = 0.2 + 0.3 = 0.5 c. Now, P(BS) is the value 0.3 in the overlapping region d. P(BS) = 0.4 + 0.3 +0.2 = 0.9. e. P(B’S’) = 1 - P(BS) = 0.1 Complement of an Event The complement of an event is the set of all outcomes that are NOT in the event. This means that if the probability of an event, A, is P(A), then the probability that the event would not occur (also called the complementary event) is 1 – P(A), denoted by P(A’). Thus, P(A’) = 1 – P(A). So the complement of an event E is the set of all the outcomes which are not in E. And together the event and its complement make all possible outcomes. Consider item e on this page. B S All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 65. D EPED C O PY 334 P(B’S’) can be determined by finding the part of the diagram where everything outside of B overlaps with everything outside of S. It is the region outside of both circles and that probability is 0.1. Another way to think of this is P(BS)’ or 1 - P(BS). The next activity will help you understand the concepts of events which are mutually exclusive and which are not mutually exclusive. As you answer the next activity, try to figure out events which are mutually exclusive and which are not. Consider the situations below and answer the questions that follow. 1. A bowl contains 15 chips numbered 1 to 15. If a chip is drawn randomly from the bowl, what is the probability that it is a. 7 or 15? b. 5 or a number divisible by 3? c. even or divisible by 3? d. a number divisible by 3 or divisible by 4? 2. Dario puts 44 marbles in a box in which 14 are red, 12 are blue, and 18 are yellow. If Dario picks one marble at random, what is the probability that he selects a red marble or a yellow marble? 3. Out of 5200 households surveyed, 2107 had a dog, 807 had a cat, and 303 had both a dog and a cat. What is the probability that a randomly selected household has a dog or a cat? Reflect: a. How did you answer each question? b. What do you notice about the events in each question? (e.g., 1.c as compared to 1. d, question 2 as compared to question 3). c. Draw a Venn diagram showing the sample space for numbers 3 and 4. What do you notice about the Venn diagrams? Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 66. D EPED C O PY 335 To help you understand the concepts of mutually exclusive and non- mutually exclusive events, read the discussion below. The events in the above activity may be either mutually exclusive or not mutually exclusive. Events that cannot occur at the same time are called mutually exclusive events. Consider the Venn diagram below. What do you notice about the events A and B? These two events are mutually exclusive. In problem 1b of the preceding activity, the event of getting a 5 and the event of getting a number divisible by 3 from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} are mutually exclusive events. If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. In symbols, P(A or B) = P(A) + P(B) On the other hand, the event of getting a number divisible by 3 or the event of getting a number divisible by 4 in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} are not mutually exclusive events. Observe that a subset of numbers divisible by 3 also contains an element which is a subset of the numbers divisible by 4. The Venn diagram below shows events A and B which are not mutually exclusive because A and B intersect. Note that there are outcomes that are common to A and B which is the intersection of A and B. P (A or B) A B All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 67. D EPED C O PY 336 If two events, A and B, are not mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities decreased by the probability of both occurring. In symbols, P(A or B)= P(A)+ P(B) – P(A and B). This section requires you to use the mathematical ideas you learned from the activity and from the discussion. Answer the problems in the following activities in different ways when possible. Consider the situation below and answer the questions that follow. 1. A restaurant serves a bowl of candies to their customers. The bowl of candies Gabriel receives has 10 chocolate candies, 8 coffee candies, and 12 caramel candies. After Gabriel chooses a candy, he eats it. Find the probability of getting candies with the indicated flavors. a. P (chocolate or coffee) c. P (coffee or caramel) b. P (caramel or not coffee) d. P (chocolate or not caramel) 2. Rhian likes to wear colored shirts. She has 15 shirts in the closet. Five of these are blue, four are in different shades of red, and the rest are of different colors. What is the probability that she will wear a blue or a red shirt? 3. Mark has pairs of pants in three different colors, blue, black, and brown. He has 5 colored shirts: a white, a red, a yellow, a blue, and a mixed-colored shirt. What is the probability that Mark wears a black pair of pants and a red shirt on a given day? 4. A motorcycle licence plate has 2 letters and 3 numbers. What is the probability that a motorcycle has a licence plate containing a double letter and an even number? Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 68. D EPED C O PY 337 Did you answer all the questions correctly? To help you understand more about mutually exclusive events, you may proceed to Activity 6. Consider each problem below. Draw a Venn diagram for each. Determine whether the events are mutually exclusive or not mutually exclusive. Then, find the probability. 1. Mario has 45 red chips, 12 blue chips, and 24 white chips. What is the probability that Mario randomly selects a red chip or a white chip? 2. Of 240 students, 176 are on the honor roll, 48 are members of the varsity team, and 36 are in the honor roll and are also members of the varsity team. What is the probability that a randomly selected student is on the honor roll or is a member of the varsity team? 3. Ruby’s dog has 8 puppies. The puppies include white females, 3 mixed-color females, 1 white male, and 2 mixed-color males. Ruby wants to keep one puppy. What is the probability that she randomly chooses a puppy that is female and white? 4. Carl’s basketball shooting records indicate that for any frame, the probability that he will score in a two-point shoot is 30%, a three-point shoot, 45%, and neither, 25%. What is the probability that Cindy will score either in a two-point shoot or in a three-point shoot? In previous lessons, you learned about counting techniques and you were able to differentiate permutation from combination. In the next activity, observe how the concepts of permutation and combination are used in solving probability problems. Activity 6: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 69. D EPED C O PY 338 Consider the situation below and answer the questions that follow. There are a total of 48 students in Grade 10 Charity. Twenty are boys and 28 are girls. 1. If a teacher randomly selects a student to represent the class in a school meeting, what is the probability that a a. boy is chosen? b. girl is chosen? 2. If a committee of 3 students is formed, what is the probability that a. all are girls? b. two are boys and one is a girl? 3. Suppose that a team of 3 students is formed such that it is composed of a team leader, a secretary, and a spokesperson. What is the probability that a team formed is composed of a girl secretary? Reflect: a. How did you answer each question? b. In finding the probability of each event above, what concepts are needed? c. Differentiate the event required in question 1 as compared to questions 2 and 3. d. Compare the events in questions 2 and 3. What necessary knowledge and skills did you need to get the correct answer? How did you compute for the probability of an event in each case? Notice that the above problems involved concepts of combination and permutation in determining the sample space and in determining the events. Do you remember your previous lesson on combination and permutation? How useful are your knowledge and skills on permutation and combination in solving problems on probability? Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 70. D EPED C O PY 339 This time you need to reflect on and further find out your understanding of compound events and mutually and not mutually exclusive events. You may also apply your knowledge and skills on the Fundamental Counting Principle, combinations, and permutations in determining the possible outcomes in the sample space. Answer the following questions 1. How does a simple event differ from a compound event? 2. Differentiate mutually exclusive events from non-mutually exclusive events. 3. Suppose there are three events A, B, and C that are not mutually exclusive. List all the probabilities you would need to consider in order to calculate P(A or B or C). Then, write the formula you would use to calculate the probability. 4. Explain why subtraction is used when finding the probability of two events that are not mutually exclusive. What new realizations do you have about probability of compound events? How would you connect this topic to other concepts that you have previously learned? How would you use this in real life? Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 71. D EPED C O PY 340 This section gives an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of probability of compound events, mutually exclusive events, and not mutually exclusive events. Answer the following questions. Write a report of your answers using a minimum of 120 words. Be ready to present your answers in the class. 1. Describe a situation in your life that involves events which are mutually exclusive or not mutually exclusive. Explain why the events are mutually exclusive or not mutually exclusive. 2. Think about your daily experience. How is probability utilized in newspapers, television shows, and radio programs that interest you? What are your general impressions of the ways in which probability is used in the print media and entertainment industry? SUMMARY/SYNTHESIS/GENERALIZATION In this lesson, you learned that an event is any collection of outcomes of an experiment. Typically, when the sample space is finite, any subset of the sample space is an event. Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events. You also learned that to each event, a probability is assigned. The probability of a compound event can be calculated if its outcomes are equally likely. Events can be mutually exclusive or not mutually exclusive. If two events, A and B, are mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities. In symbols, P(A or B) = P(A) + P(B). On the other hand, if two events, A and B, are not mutually exclusive, then the probability that either A or B occurs is the sum of their probabilities decreased by the probability of both of them occurring. In symbols, P (A or B) = P (A) + P (B) – P (A and B). Activity 9: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 72. D EPED C O PY 341 In Lesson 1 of this module, you learned about the basic concepts of the probability of compound events. You will formally learn, for example, why the outcome of the flip of a fair coin is independent of the flips that came before it. Consider the situations below and answer the questions that follow. Situation 1: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. A ball is drawn at random and the color is noted and then put back inside the box. Then, another ball is drawn at random. Find the probability that: a. both are blue. b. the first is red and the second is yellow. Situation 2: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. Suppose that two balls are drawn one after the other without putting back the first ball. Find the probability that: a. the first is red and the second is blue. b. both balls are yellow. Reflect: a. Compare the process of getting the probabilities in each of the situations above? b. In situation 1, is the probability of obtaining the second ball affected by the first ball? What about in situation 2? c. What conclusion can you make about events happening in the given situations above? How are these events different? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 73. D EPED C O PY 342 You have just completed Activity 1 on understanding dependent and independent events. Continue by considering the discussion that follows. Please read thoroughly and if needed, you may go back to the situations given in Activity 1 of this lesson. Independent and Dependent Events In situation 1, the probability of getting a blue ball in the second draw is not affected by the probability of drawing a red ball on the first draw, since the first ball is put back inside the box prior to the second draw. Thus, the two events are independent of each other. Two events are independent if the outcome of one event does not affect the outcome of the other event. Example: When a coin is tossed and a die is rolled, the event that a coin shows up head and the event that a die shows up a 5 are independent events. Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other. If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B) = P(A) • P(B) When the outcome of one event affects the outcome of another event, they are dependent events. In situation 2, if the ball was not placed back in the box, then drawing the two balls would have been dependent events. In this case, the event of drawing a yellow ball on the second draw is dependent on the event of drawing a yellow ball on the first draw. Example: A box contains 7 white marbles and 7 red marbles. What is the probability of drawing 2 white marbles and 1 red marble in succession without replacement? On the first draw, the probability of getting a white marble is 7 14 . On the second draw, the probability of getting a white marble is 6 13 . Then on the third draw, the probability of getting a red marble is 7 12 . So, P(1 white, 1 white, 1 red) = 7 6 7 7 14 13 12 52    All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 74. D EPED C O PY 343 If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) = P(A) • P(B following A) This section requires you to use the mathematical ideas you learned from the activity and from the discussion. Answer the problems in the following activities in different ways when possible. Consider the situation below and answer the questions that follow. 1. A bag contains 6 black marbles, 9 blue marbles, 4 yellow marbles, and 2 green marbles. A marble is randomly selected, replaced, and a second marble is randomly selected. Find the probability of selecting a black marble, then a yellow marble. 2. A box of chocolates contains 10 milk chocolates, 8 dark chocolates, and 6 white chocolates. Hanissa randomly chooses a chocolate, eats it, and then randomly chooses another chocolate. What is the probability that Hanissa chose a milk chocolate, and then, a white chocolate? 3. A rental agency has 12 white cars, 8 gray cars, 6 red cars, and 3 green cars for rent. Mr. Escobar rents a car, returns it because the radio is broken, and gets another car. What is the probability that Mr. Escobar is given a green car and then a gray car? Did you answer all the questions correctly? If not, you may go back to the discussion. Otherwise you may proceed to Activity 3. Now, using your own words, differentiate independent events from dependent events. Then, you may answer the questions in the next activity. Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 75. D EPED C O PY 344 Determine whether the events are independent or dependent. Then, find the probability. 1. A toy box contains 12 toys, 8 stuffed animals, and 3 board games. Maria randomly chooses 2 toys for the child she is babysitting to play with. What is the probability that she chose 2 stuffed animals as the first two choices? 2. A basket contains 6 apples, 5 bananas, 4 oranges, and 5 guavas. Dominic randomly chooses one piece of fruit, eats it, and chooses another piece of fruit. What is the probability that he chose a banana and then an apple? 3. Nick has 4 black pens, 3 blue pens, and 2 red pens in his school bag. Nick randomly picks two pens out of his school bag. What is the probability that Nick chose two blue pens, if he replaced the first pen back in his pocket before choosing a second pen? This time you need to reflect on and further find out your understanding of dependent and independent events. Consider the situation below and answer the questions that follow. 1. A bag of jelly beans contains 10 red, 6 green, 7 yellow, and 5 orange jelly beans. What is the probability of randomly choosing a red jelly bean, replacing it, randomly choosing another red jelly bean, replacing it, and then randomly choosing an orange jelly bean? 2. Rene and Cris went to a grocery store to buy drinks. They chose from 10 different brands of juice drinks, 6 different brands of carbonated drinks, and 3 different brands of mineral water. What is the probability that Rene and Cris both chose juice drinks, if Rene randomly chose first and liked the first brand he picked up? Activity 4: Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 76. D EPED C O PY 345 Answer the following questions. 1. What makes two events independent? 2. Differentiate a dependent event from an independent event. Reflect: a. What new realizations do you have about the probability of a dependent event? b. How would you connect this topic to other concepts that you have previously learned? c. How would you use these concepts in real life? This section is an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of probability of compound events, dependent events, and independent events. 1. Describe a situation in your life that involves dependent and independent events. Explain why the events are dependent or independent. 2. Formulate your own problems involving independent and dependent events. SUMMARY/SYNTHESIS/GENERALIZATION Two events are independent if the occurrence of one of the events gives no information about whether or not the other event will occur; that is, the events have no influence on each other. If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B) = P(A) • P(B). When the outcome of one event affects the outcome of another event, they are dependent events. If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) = P(A) • P(B following A). Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 77. D EPED C O PY 346 Conditional probability plays a key role in many practical applications of probability. In these applications, important conditional probabilities are often drastically affected by seemingly small changes in the basic information from which the probabilities are derived. In this lesson, we will focus on finding a particular kind of probability called a conditional probability. To understand conditional probability, you may begin by answering Activity 1. Consider the situation and answer the questions that follow. Mario bought four different batteries. Of these four, one is defective. Two are to be selected at random for use on a particular day. 1. Draw a tree diagram associated with the experiment of selecting two batteries from among four, in which one is defective. 2. List the sample space. How many outcomes are there? 3. Find the probability that the second battery selected is not defective. 4. What if you find the probability that the second battery selected is not defective, given that the first was not defective? Reflect: a. How did you answer question number 3? How is the condition different from that of question number 4? b. How did you find the probability that the second battery selected was not defective, given that the first was not defective? In Activity 1, you notice that a condition was given when you were asked to find the probability of an event. How does the given condition affect the probability of an event? Activity 1 shows an example of probability involving conditions which we refer to as conditional probability. To understand conditional probability further, you may proceed to Activity 2. Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 78. D EPED C O PY 347 Consider the situation and answer the questions that follow. A proficiency examination for certain technical skills was given to 100 employees of a manufacturing firm. The breakdown of the examination results of the employees are shown in the table below. Male (M) Female (F) Total Pass (P) 24 36 60 Fail (Q ) 16 24 40 40 60 100 Suppose that an employee is selected at random from among the 100 employees who took the examination. 1. Are the events P and M independent? Explain. 2. Are the events P and F independent? Explain. 3. Find the probability that the employee passed the exam, given that he was a male. 4. Find the probability that the employee was a male, given that a passing grade was received. 5. Find the probability that the employee was a female, given that a passing grade was received. In many situations, once more information become available, we are able to revise our estimates for the probability of further outcomes or events happening. In this lesson, we are interested in answering this type of question: how the information “an event B has occurred" affects the probability that “event A occurs." The usual notation for "event A occurs given that event B has occurred" is "A|B" (A given B). The symbol | is a vertical line and does not imply division. P(A|B) denotes the probability that event A will occur given that event B has occurred already. We define conditional probability as follows: For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by ( ) ( | ) ( ) P A B P A B P B   Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 79. D EPED C O PY 348 When two events, A and B, are dependent, the probability of both events occurring is P(A and B) = P(B) P(A|B). Also, P(A and B) = P(A) P(B|A). Sample Problem: A mathematics teacher gave her class two tests. Twenty- five percent of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test? Solution: This problem involves a conditional probability since it asks for the probability that the second test was passed given that the first test was passed.       P First and Second P Second|First P First  0.25 0.42  25 42   0.60 or 60% Consider the table below showing A as the age group under 30 years old who purchase 2 different brands of shoes. Age Group Brand X Brand Y Total A (under 30 years old) 6% 34% 40% A’ (30 years and older) 9% 51% 60% Total 15% 85% 100% 1. What is the probability that a person chosen at random purchases Brand X? 2. What is the probability that a person chosen at random is under 30 years old? 3. What is the probability that a person chosen at random purchases Brand X and is under 30 years old? 4. What is the probability that a person chosen at random purchases Brand X and he or she is under 30 years old? Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 80. D EPED C O PY 349 Question for discussion: Does the occurrence of event A give information about the probability of event X? Why or Why not? To understand further conditional probability, study the solution to the above problem. Solution: We use a Venn diagram to show the relationships of the events. 1. P(X) = 0.15 2. P(A) = 0.40 3. P X A( )= 0.06 4.     ( ) | P P X P X A A A   . . 0 06 0 40  .0 15 Take note that from 1) and 4), P(X) and P(X|A) are both equal to 0.15. Notice that the occurrence of event A gives no information about the probability of event X. The events X and A are independent events. Two events A and B are said to be independent if either: i. P(A | B) = P(A), i.e., P(B | A) = P(B), or equivalently, ii. P(A ∩ B) = P(A) P(B). Probabilities are usually very sensitive to an information given as a condition. Sometimes, however, a probability does not change when a condition is supplied. If the extra information provided by knowing that an event B has occurred does not change the probability of A, that is, if, ( | ) ( )P A B P A then events A and B are said to be independent. Since     ( ) | P P A P A B B B   Notice that two events A and B are said to be independent if P(A B) = P(A) P(B). This is equivalent to stating that P(A|B) = P(A) and P(B|A) = P(B) if these probabilities exist. 0.06 0.51 0.09 S X A 0.34 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 81. D EPED C O PY 350 Sometimes a conditional probability is known, and we want to find the probability of an intersection. By rearranging the terms in the definition of conditional probability and considering the definition of independence, we obtain the Multiplicative Rule which is P(A ∩ B) = P(A)P(B). Let us look at some other problems in which you are asked to find a conditional probability in Activity 4. This section requires you to use the mathematical ideas you learned from the previous activities and from the discussion. Answer the problems in the following activities in different ways when possible. Answer the following questions. 1. Given that P(S) = 0.3, P(S|Q) = 0.4, and P(Q) = 0.5, find the following probabilities: a. P(SQ) b. P(Q|S) c. P(S’|Q) d. P(S|Q’) 2. Assume S and Q are independent events with P(S) = 0.2 and P(Q) = 0.3. Let T be the event that at least one of S or Q occurs, and let R be the event that exactly one of S or Q occurs. Find the following probabilities a. P(T) b. P(R) c. P(S|R) d. P(R|S). e. Determine whether S and R are independent. Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 82. D EPED C O PY 351 Reflect: a. What do you notice about the conditional probability of independent events? b. How about the conditional probability of dependent events? 1. A family has two children. What is the probability that the younger child is a girl, given that at least one of the children is a girl? 2. At a basketball game, 80% of the fans cheered for team A. In the same crowd, 20% of the fans were waving banners and cheering for team A. What is the probability that a fan waved a banner given that the fan cheered for team A? This time you need to reflect on and further find out your understanding of dependent and independent events. For numbers 1 to 3, consider the Venn diagram on the right. 1. What does the Venn diagram illustrate? 2. Using the Venn diagram, how do you find P(B|A)? 3. Write a situation that can be illustrated by the Venn diagram. Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 83. D EPED C O PY 352 This section provides an opportunity for you to apply what you have learned in this lesson to real-life situations. You are asked to do certain tasks that will demonstrate your understanding of conditional probability. Make a research report. Choose your own topic of study or choose from any of the four recommended topics given below. Focus on the question that follows: How can I use statistics and probability to help others make informed decisions regarding my chosen topic? Recommended Topics: 1. Driving and cell phone use 2. Diet and health 3. Professional athletics 4. Costs associated with a college education SUMMARY/SYNTHESIS/GENERALIZATION For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by     ( ) | P P A P A B B B   In word problems, conditional probabilities can usually be recognized by words like “given”, “if,” or “among” (e.g., in the context of samples). There are, however, no hard rules, and you have to read the problem carefully and pay attention to the entire context of the problem to determine whether the given probability represents an ordinary probability (e.g., P(AB)) or a conditional probability (e.g., P(A|B) or P(B|A)). Activity 7: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 84. D EPED C O PY 353 GLOSSARY OF TERMS Complement of an Event – a set of all outcomes that are NOT in the event. If A is the event, the complement of the event A is denoted by A’ Compound Events – a composition of two or more simple events Conditional Probability - The conditional probability of an event B given A is the probability that the event B will occur given that an event A has already occurred. This probability is written as P(B|A) and read as the probability of B given A. In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is, P(B). Dependent Events – Two events are dependent if the occurrence of one event does affect the occurrence of the other (e.g., random selection without replacement). Events – a set of possible outcomes resulting from a particular experiment. For example, a possible event when a single six-sided die is rolled is {5, 6}, that is, the roll could be a 5 or a 6. In general, an event is any subset of a sample space (including the possibility of an empty set). Independent Events – events in which the probability of any one event occurring is unaffected by the occurrence or non-occurrence of any of the other events. Formally, A and B are independent if and only if P(A|B) = P(A). Intersection of Events – a set that contains all of the elements that are in both events. The intersection of events A and B is written as A B . Mutually Exclusive Events – events that have no outcomes in common. This also means that if two or more events are mutually exclusive, they cannot happen at the same time. This is also referred to as disjoint events. Union of Events – a set that contains all of the elements that are in at least one of the two events. The union is written as .A B Venn Diagram – a diagram that uses circles to represent sets, in which the relations between the sets are indicated by the arrangement of the circles. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  • 85. D EPED C O PY 354 REFERENCES AND WEBSITE LINKS USED IN THIS MODULE References: Canonigo, A. (2000). Historical, Philosophical and Psychological Considerations in Probability Theory Intersection: Official Journal of Philippine Council of Mathematics Teacher Educators (MATHED), Inc. Lee-Chua, Editor. MATHED, Inc: Quezon City. 1(2), 22-32. Carpenter, T. Corbitt, M., Kepner, H., Lindquist, M. & Reys, R. (1981). “What are the chances of your students knowing probability?” The Mathematics Teacher 74(5), 342-344. Website Links as Reference and Sources of Learning Activities: Conditional Probability and Independence. Tutorials for Finite Math. http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/frames6_5.html Accessed on 3 March 2014. Conditional Probability. Wolfram MathWorld. http://mathworld.wolfram.com/ConditionalProbability.html Accessed on 4 March 2014. Conditional Probability: Definitions and non-trivial examples. Cut the Knot. http://www.cut-the-knot.org/Probability/ConditionalProbability.shtmlAccessed on 6 March 2014. Conditional Probability.http://www.stat.yale.edu/Courses/1997- 98/101/condprob.htm Accessed on 6 March 2014 Independent Events. Math Goodies. http://www.mathgoodies.com/lessons/vol6/independent_events.html Accessed on 6 March 2014. Mutually Exclusive Events. Math Is Fun. http://www.mathsisfun.com/data/probability-events-mutually-exclusive.html Accessed on 6 March 2014 The Monty Hall Dilemma. Cut the Knot .http://www.cut-the-knot.org/hall.shtml Accessed on 6 March 2014 The Monty Hall Problem.http://montyhallproblem.com/ Accessed on 6 March 2014 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.