SECTION 4.2 Binomial Distributions 203 Binomial Probability Formula There are several ways to find the probability of x successes in n trials of a binomial experiment. One way is to use a tree diagram and the Multiplication Rule. Another way is to use the binomial probability formula. In a binomial experiment, the probability of exactly x successes in n trials is P1x2 = nCxp xqn-x = n! 1n - x2! x! pxqn-x. Note that the number of failures is n - x. Binomial Probability Formula Finding a Binomial Probability Rotator cuff surgery has a 90% chance of success. The surgery is performed on three patients. Find the probability of the surgery being successful on exactly two patients. (Source: The Orthopedic Center of St. Louis) SOLUTION Method 1: Draw a tree diagram and use the Multiplication Rule. 3rd Surgery 2nd Surgery 1st Surgery S F S F S F S F 3 2 2 1 2 1 1 0 SSS SSF SFS SFF FSS FSF FFS FFF Probability Outcome Number of Successes 9 10 9 10 9 10 729 1000 . . = 9 10 9 10 1 10 81 1000 . . = 9 10 1 10 9 10 81 1000 . . = 9 10 1 10 1 10 9 1000 . . = 1 10 9 10 9 10 81 1000 . . = 1 10 9 10 1 10 9 1000 . . = 1 10 1 10 9 10 9 1000 . . = 1 10 1 10 1 10 1 1000 . . = S S F S F F There are three outcomes that have exactly two successes, and each has a probability of 81 1000. So, the probability of a successful surgery on exactly two patients is 31 81 10002 = 0.243. Method 2: Use the binomial probability formula. In this binomial experiment, the values of n, p, q, and x are n = 3, p = 9 10 , q = 1 10 , and x = 2. The probability of exactly two successful surgeries is P122 = 3! 13 - 22!2!a 9 10b 2a 1 10b 1 = 3a 81 100ba 1 10b = 3a 81 1000b = 0.243. TRY IT YOURSELF 2 A card is selected from a standard deck and replaced. This experiment is repeated a total of five times. Find the probability of selecting exactly three clubs. Answer: Page A38 EXAMPLE 2 Study Tip In the binomial probability formula, nCx determines the number of ways of getting x successes in n trials, regardless of order. nCx = n! 1n - x2!x! Study Tip Recall that n! is read “n factorial” and represents the product of all integers from n to 1. For instance, 5! = 5# 4# 3# 2# 1 = 120.
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