1106 CHAPTER 11 Further Topics in Algebra Suppose that we want to determine the probability of getting exactly 3 tails in 5 coin tosses. Here n = 5, r = 3, p = P1tail2 = 1 2, and q = P1head2 = 1 - 1 2 = 1 2. P13 tails in 5 coin tosses2 = a 5 3ba 1 2b 3a1 2b 5-3 Use the binomial probability formula. = 5! 3!2! a 1 2b 3 a1 2b 2 Apply the formula for C1n, r2 = Q n rR and subtract. = 10a 1 2b 3 a1 2b 2 Evaluate factorials and divide. = 0.3125 Apply the exponents and multiply. ALGEBRAIC SOLUTION (a) There are n = 10 independent trials with p = P132 = 1 6 and q = 1 - 1 6 = 5 6. P14 threes in 10 rolls2 = a 10 4 ba 1 6b 4a5 6b 10-4 = 210a 1 6b 4a5 6b 6 ≈0.054 (b) Here n = 10, p = P1not a 32 = 5 6, and q = 1 6. P19 non@threes in 10 rolls2 = a 10 9 ba 5 6b 9a1 6b 1 ≈0.323 GRAPHING CALCULATOR SOLUTION Graphing calculators, such as the TI-84 Plus, that have statistical distribution functions give binomial probabilities. Figure 19 shows the results for parts (a) and (b). The numbers in parentheses separated by commas represent n, p, and r, respectively. S Now Try Exercise 41. EXAMPLE 6 Finding Probabilities in a Binomial Experiment An experiment consists of rolling a fair die 10 times and observing the number of 3s. (a) Find the probability of getting exactly 4 threes. (b) Find the probability that the result is not a 3 in exactly 9 of the rolls. Figure 19 CONCEPT PREVIEWFill in the blank(s) to correctly complete each sentence. 1. When a fair coin is tossed, there are possible outcomes, and the probability of each outcome is . 2. When a fair die is rolled, there are possible outcomes, and the probability of each outcome is . 11.7 Exercises
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