5-2 Binomial Probability Distributions 221 Method 1: Using the Binomial Probability Formula In a binomial probability distribution, probabilities can be calculated by using Formula 5-5. FORMULA 5-5 Binomial Probability Formula P1x2 = n! 1n - x2!x! # px # qn-x for x = 0, 1, 2, c, n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial 1q = 1 - p2 Fun Stuff for Formula-Loving Readers Formula 5-5 can also be expressed as P1x2 = nCx p x qn-x. With x items identical to themselves, and n - x other items identical to themselves, the number of permutations is nCx = n!>31n - x2!x!4, so the two sides of this equation are interchangeable. The factorial symbol !, introduced in Section 4-4, denotes the product of decreasing factors. Two examples of factorials are 3! = 3 # 2# 1 = 6 and 0! = 1 (by definition). Cash EXAMPLE 2 Given that there is a 0.05 probability that a randomly selected adult smartphone owner is cashless, use the binomial probability formula to calculate the probability that when ten adults are randomly selected, exactly two of them are cashless. That is, apply the binomial probability formula (Formula 5-5) to find P122 given that n = 10, x = 2, p = 0.05, and q = 0.95. Using the given values of n, x, p, and q in the binomial probability formula (Formula 5-5), we get P122 = 10! 110 - 22!2! # 0.052 # 0.9510-2 = 10! 8!2! # 0.0025# 0.663420 = 145210.0025210.6634202 = 0.074635 = 0.0746 1rounded to three significant digits2 The probability of getting exactly two cashless adults is 0.0746. SOLUTION YOUR TURN. Do Exercise 13 “Guessing Answers.” Calculation hint: When computing a probability with the binomial probability formula, it’s helpful to get a single number for n!>31n - x2!x!4 or nCx, a single number for px, and a single number for qn-x, then simply multiply the three factors together as shown in the third line of the calculation in the preceding example. Don’t round when you find those three factors; round only at the end, and round the final result to three significant digits.
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