Basic Counting Law Discrete Math Power Point Presentaton
- 2. Section Summary What is Probability? Basic Counting Rule
- 3. Experiment and Its Sample Space An experiment (a real experiment or a thought experiment) is any process that generates well-defined outcomes The set of all possible outcomes is called sample space usually denoted by 𝑆. An experimental outcome is also called a sample point. A subset of sample space is called an event Examples: Flip a coin Weather in Naryn KG on February 14, 2035 Toss a die or a pair of dice Choose a card from a standard deck of cards Temperature in Gulakain Khorog at 5 pm today
- 4. Example Tossing a die. The sample space is S = {1, 2, 3, 4, 5, 6}. Give an example of an event in this experiment. E = {2, 4, 6} is an event, which can be described in words as ”the number is even”
- 5. What is probability? Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near zero indicates an event is quite unlikely to occur. A probability near one indicates an event is almost certain to occur
- 6. Example If I have 3 red marbles and 5 blue marbles in a bag, what is the probability (or chance, or likelihood) that it is red if I pull out one marble without looking in the bag?
- 7. Dice Problem P (red) = ------- Probability of the event of pulling out a red marble 3 8 Number of red marbles - or number of outcomes that result in the event Total number of marbles - or total number of possible outcomes This entire thought process is what we call in probability a ‘thought experiment’, or just an ‘experiment’
- 8. Dice Problem Try: For the experiment of rolling a single six- sided die, find the probability of rolling: i) 5 ii) 7 iii) a positive integer
- 9. Eg., If 47% of students in the UCA are female, and I randomly select a student from all the UCA students, what is the probability that the student is male?
- 10. Try: A question on a multiple-choice test has five possible answers, only one of which is correct. If you guess on the question, what is the probability of guessing an incorrect answer?
- 11. Try: An experiment consists of tossing a coin three times in a row. a) List all possible outcomes (sample space). b) What is the probability of getting one head? c) What is Probability of getting two heads? d) What is Probability of getting three heads?
- 12. Try: An experiment consists of tossing a coin three times in a row. a) List all possible outcomes. S={HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (8 possible outcomes)
- 13. How many outcomes are there for the experiment of flipping a coin and throwing a regular die? H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 So, 12 possible outcomes. There are ‘2 possible ways the first task can be completed’ (ie., either H or T) There are ‘6 possible ways the second task can be completed’ (ie., 1, 2, 3, 4, 5 or 6) How does the 12 relate to the 2 and 6? Basic Counting Law 2 x 6 =12
- 14. Case 1 Try: For lunch you have 5 choices of salad and 7 choices for an entree. How many different lunch possibilities are there? 5 x 7 = 35 We can extend the Basic Counting Law to more than two tasks. Case 2 Try: For lunch you have 5 choices of salad, 7 choices for an entree and 3 choices for dessert. How many different lunch possibilities are there now? 5 x 7 x 3 = 105 Basic Counting Law
- 15. Case 3 Try: A die is rolled four times in a row. How many outcomes are possible? 6 x 6 x 6 x 6 = 1296 (there are four tasks here) Basic Counting Law
- 16. Example Suppose you own 5 shirts and 3 pants. How many ways are there for you to get dressed in the morning?
- 17. In general… Basic Counting Law: If there are m possible ways a task can be performed and, after the first task is complete, there are n possible ways for a second task to be performed, then there are m • n possible ways for the two tasks to be performed in order. Basic Counting Law
- 18. On using the Basic Counting Law
- 19. Players’ shirts have two digit numbers. Ten possible digits: 0,1,2,3,4,5,6,7,8,9 You want to choose a two digit number for your shirt. How many possible two digit numbers can you choose from? 10 × 10 = 100 If you can not repeat the same digit, how many choices you have? 10 × 9 = 90 Player jersey Numbers
- 20. License plates
- 21. Example How many different license plates can be made if each plate contains a sequence of three uppercase English letters followed by three digits?
- 22. Solution Example: How many different license plates can be made if each plate contains a sequence of three uppercase English letters followed by three digits? Solution: By the product rule, there are 26 ∙ 26 ∙ 26 ∙ 10 ∙ 10 ∙ 10 = 17,576,000 different possible license plates.
- 23. E.g. Suppose a license plate consists of a letter, followed by four numbers, followed by another letter. How many different license plates are possible if: a) any letter and digits are allowed? b) no repetition of letters or digits is allowed? c) it must start with “B”? License plate problem
- 24. How many different license plates are possible if: a) any letter and digits are allowed? 26 x 10 x 10 x 10 x 10 x 26 = 6 760 000 (there are 6 tasks in these questions) b) no repetition of letters or digits is allowed? 26 x 10 x 9 x 8 x 7 x 25 = 3 276 000 c) it must start with “B”? 1 x 10 x 10 x 10 x 10 x 26 = 260 000 Solution
- 25. Exercise What is the number of total outcomes if a 6- sided die is rolled, a. 2 times b. 3 times c. 4 times d. 𝑘 times