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Chapter 3 Probability § 3.4 Counting Principles Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m· n. This rule can be extended for any number of events occurring in a sequence. Example: A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available? # of main # of side dishes dishes 4 2 There are 40 meals available. # of desserts 5 Larson & Farber, Elementary Statistics: Picturing the World, 3e = 40 3 Fundamental Counting Principle Example: Two coins are flipped. How many different outcomes are there? List the sample space. Start 1st Coin Tossed Heads 2 ways to flip the coin Tails 2nd Coin Tossed Heads Tails Heads Tails 2 ways to flip the coin There are 2 2 = 4 different outcomes: {HH, HT, TH, TT}. Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Fundamental Counting Principle Example: The access code to a house's security system consists of 5 digits. Each digit can be 0 through 9. How many different codes are available if a.) each digit can be repeated? b.) each digit can only be used once and not repeated? a.) Because each digit can be repeated, there are 10 choices for each of the 5 digits. 10 · 10 · 10 · 10 · 10 = 100,000 codes b.) Because each digit cannot be repeated, there are 10 choices for the first digit, 9 choices left for the second digit, 8 for the third, 7 for the fourth and 6 for the fifth. 10 · 9 · 8 · 7 · 6 = 30,240 codes Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Permutations A permutation is an ordered arrangement of objects. The n umber of different permutations of n distinct objects is n!. “n factorial” n! = n · (n – 1)· (n – 2)· (n – 3)· …· 3· 2· 1 Example: How many different surveys are required to cover all possible question arrangements if there are 7 questions in a survey? 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 surveys Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Permutation of n Objects Taken r at a Time The number of permutations of n elements taken r at a time is n Pr # in the group n! . (n r)! # taken from the group Example: You are required to read 5 books from a list of 8. In how many different orders can you do so? n Pr 8 P5 8! 8! = 8 7 6 5 4 3 2 1 6720 ways (8 5)! 3! 3 2 1 Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 Distinguishable Permutations The number of distinguishable permutations of n objects, where n1 are one type, n2 are another type, and so on is n! , where n1 n2 n3 nk n. n1 ! n2 ! n3 ! nk ! Example: Jessie wants to plant 10 plants along the sidewalk in her front yard. She has 3 rose bushes, 4 daffodils, and 3 lilies. In how many distinguishable ways can the plants be arranged? 10 9 8 7 6 5 4! 10! 3!4!3! 3!4!3! 4,200 different ways to arrange the plants Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Combination of n Objects Taken r at a Time A combination is a selection of r objects from a group of n things when order does not matter. The number of combinations of r objects selected from a group of n objects is # in the collection nC r n! . (n r)! r ! # taken from the collection Example: You are required to read 5 books from a list of 8. In how many different ways can you do so if the order doesn’t matter? 8! = 8 7 6 5! 8C 5 = 3!5! 3!5! = 56 combinations Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 Application of Counting Principles Example: In a state lottery, you must correctly select 6 numbers (in any order) out of 44 to win the grand prize. a.) How many ways can 6 numbers be chosen from the 44 numbers? b.) If you purchase one lottery ticket, what is the probability of winning the top prize? a.) 44! 7,059,052 combinations C 44 6 6!38! b.) There is only one winning ticket, therefore, 1 P (win) 0.00000014 7059052 Larson & Farber, Elementary Statistics: Picturing the World, 3e 10