Small Business Owners Seventy-six percent of small business owners do not have a college degree. If a random sample of 60 small business owners is selected, find the probability that exactly 47 will not have a college degree. Round the final answer to at least 4 decimal places and intermediate \( z \)-value calculations to 2 decimal places. \( P(X=47)= \)
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To solve this problem, we can use the binomial probability formula, which is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( n \) = total number of trials (in this case, 60 small business owners) - \( k \) = number of successful trials (in this case, 47 small business owners without a college degree) - \( p \) = probability of success on any given trial (in this case, 0.76) - \( \binom{n}{k} \) = the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \) For our problem: - \( n = 60 \) - \( k = 47 \) - \( p = 0.76 \) First, we calculate the binomial coefficient \( \binom{60}{47} \): \[ \binom{60}{47} = \frac{60!}{47! \cdot (60-47)!} = \frac{60!}{47! \cdot 13!} \] Now, calculating \( P(X = 47) \): 1. \( \binom{60}{47} = \binom{60}{13} \) (since \( \binom{n}{k} = \binom{n}{n-k} \)), and computing \( \binom{60}{13} \) gives a large number, specifically 10,442,452,300. 2. Substitute into the probability formula: \[ P(X = 47) = \binom{60}{47} (0.76)^{47} (0.24)^{13} \] First we calculate \( (0.76)^{47} \approx 2.7406 \times 10^{-6} \) and \( (0.24)^{13} \approx 5.4923 \times 10^{-6} \). Now calculate \( P(X = 47) \): \[ P(X = 47) = 10,442,452,300 \cdot (2.7406 \times 10^{-6}) \cdot (5.4923 \times 10^{-6}) \] \[ \approx 10,442,452,300 \cdot 1.5027 \times 10^{-11} \] \[ \approx 0.1570 \] Finally, round to four decimal places: \[ P(X = 47) \approx 0.1570 \] Thus, the probability that exactly 47 out of 60 small business owners do not have a college degree is approximately **0.1570** or **15.70%**.
