Counting techniques and probability
- 2. If the saying “ nothing is certain except death and taxes” is true, then all of the other events in our lives are uncertain. For instance, will it rain today? Will you pass the exam? Will the package arrive on time? Will the plane land safely? Will our team win the game? All of these are uncertain events and the study of uncertain events is called probability. Finding the probability of an event requires counting the elements of the sample space. Sometimes sample space is so large that shortcuts are needed to count all the possibilities. Hence, we need to learn useful counting techniques such as fundamental principle of counting, permutation, and combination What is the chance of getting struck by a lightning, winning a lotto, or getting a girl you love say “YES”?
- 3. Tree diagrams and the Fundamental Principle of Counting 1.Sarah Geronimo is a popular concert artist. Suppose she is planning a concert tour in three cities – Manila, Cebu , and Davao. In how many ways can she arrange her tour schedule?If there is no restriction on the order of performances, then Sarah may start in one of the cities. After the first city is chosen, she may choose from the two remaining cities as her second stop. The remaining city is her last stop. The tree diagram below shows the possible tour schedules, First Stop. Second stop. Third Stop Manila Cebu. ---------- Davao Davao --------------- 6 possible outcomes
- 4. 2.A FOOD STALL SELLS SQUID BALL, FISH BALL, AND KIKIAM. THERE ARE ALSO THREE CHOICES FOR THE SAUCE: SWEET, SPICY OR SWEET AND SPICY. HOW MANY DIFFERENT COMBINATIONS ARE POSSIBLE?
- 5. ■ 2. Food Choice Sauce Outcomes ■ squid ball sweet _________ SB , sweet ■ ( SB) spicy _________ SB , spicy ■ sweet n spicy _______SB , aweet n spicy ■ START Fish ball sweet __________FB , sweet ■ (FB) spicy __________FB , spicy ■ sweet n spicy_________FB , swet n spicy ■ kikiam sweet ________K , sweet ■ (K) spicy _______ K , spicy ■ Sweet n spicy _______ K , sweet n spicy there are 9 possible outcomes
- 6. ■ 3. A coin is tossed thrice. Draw a tree diagram to illustrate the possible outcomes. ■ First toss second toss third toss outcomes ■ H H HHH ■ H T T HHT ■ Start H ____ HTH ■ T _____ HTT ■ T H H _____THH ■ T _____THT ■ T H _____TTH ■ T _____TTT there are 8 possible outcomes
- 7. WHEN ATASK CAN BE DONE INTWO OR MORE STAGES AND EACH STAGE CAN BE DONE IN A NUMBER OF WAYS , TREE DIAGRAM HELP IN SHOWINGTHE POSSIBLE CHOICES AND IN DETERMININGTHE NUMBER OF WAYS THATTHEWHOLETASK CAN BE DONE. , WhenTHE COMBINATION OF ITEMS OR A SUCCESSION OF EVENTS ARE CONSIDERED, EACH RESULT IS CALLED AN OUTCOME. AN EVENT IS SUBSET OF ALL POSSIBLE OUTCOME. A COMPOUND EVENT OCCURS WHEN AN EVENT IS COMPOSED OFTWO OR MORE OUTCOMES, SUCH AS FLIPPING A COIN FOLLOWED BY FLIPPING ANOTHER COIN.
- 8. GENERALIZATION: ■ THE FUNDAMENTAL PRINCIPLE OF COUNTING ■ In a compound event in which the first event may occur in n1 different ways, the second event may occur in n2 different ways, and so on and the kth event may occur in nk different ways, the total number of ways the compound event may occur is n1 * n2 * n3 ...... nk . ■ Ex 1. ConcertTour Schedule ■ First Stop Second Stop Third Stop Number of PossibleWays ■ 3 2 1 6 ■ 2. Order in Food Stall ■ Food Choice Sauce Number of possibleWays ■ 3 3 9 ■ 3.Tossing a CoinThrice ■ 1stToss 2ndToss 3rd toss number of PossibleWays ■ 2 2 2 8
- 9. PRACTICE EXERCISES: ■ TRYTHIS ■ 1. In a cafe , coffee can be ordered in three sizes, small (S) medium (M) , and ■ large (L), and in four flavors - Cappuccino (C) , Machiatto (M) Americano ( A) , and ■ Latte (L). In how many ways can you order coffee? ■ 2. Suppose an experiment consist of tossing a coin four times. ■ a. How many different outcomes are possible? ■ b. how many outcomes have at least 1 head? ■ ■ c. How many outcomes have at most 2 heads? ■ d. How many different outcomes have exactly 3 heads 16 15, all outcomes,exceptTTTT have at least one head 1 + 4 + 6 = 11 4 3 x 4 = 12 Hint:The outcomes with outmost 2 heads corresponds to the sequence with 0 , 1 or 2 heads
- 10. ASSIGNMENT: ■ “HAPPY BIRTHDAYTOYOU” ■ Andy asked his girlfriend to make all the decisions for their date on her birthday. She will pick a restaurant and an activity for the date.Andy will choose a gift for her.The local restaurants include Korean , Chinese , and Italian.The activities she can choose from are bowling, movies and putt-putt. Andy will buy either chocolate or flowers. ■ 1. How many outcomes are there for these three decisions? ■ 2. Draw a tree diagram to illustrate the choices.