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. \( P(X=47)= \)

To find the probability that exactly 47 small business owners out of 60 do not have a college degree, we can use the binomial distribution formula. Given: - \( n = 60 \) (total number of small business owners) - \( p = 0.76 \) (probability of a small business owner not having a college degree) - \( X = 47 \) (number of small business owners not having a college degree) The binomial distribution formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substitute the given values into the formula: \[ P(X = 47) = \binom{60}{47} (0.76)^{47} (1-0.76)^{60-47} \] Now, we can calculate the probability using the binomial distribution formula. Calculate the value by following steps: - step0: Calculate: \( { }_{60}C_{47}\times 0.76^{47}\left(1-0.76\right)^{60-47}\) - step1: Subtract the numbers: \( { }_{60}C_{47}\times 0.76^{47}\times 0.24^{60-47}\) - step2: Subtract the numbers: \( { }_{60}C_{47}\times 0.76^{47}\times 0.24^{13}\) - step3: Expand the expression: \(\frac{60!}{47!\times \left(60-47\right)!}\times 0.76^{47}\times 0.24^{13}\) - step4: Subtract the numbers: \(\frac{60!}{47!\times 13!}\times 0.76^{47}\times 0.24^{13}\) - step5: Multiply the terms: \(\frac{60!}{129729600\times 48!}\times 0.76^{47}\times 0.24^{13}\) - step6: Divide the terms: \(5166863427600\times 0.76^{47}\times 0.24^{13}\) - step7: Convert the expressions: \(5166863427600\left(\frac{19}{25}\right)^{47}\times 0.24^{13}\) - step8: Convert the expressions: \(5166863427600\left(\frac{19}{25}\right)^{47}\left(\frac{6}{25}\right)^{13}\) - step9: Simplify: \(5166863427600\times \frac{19^{47}}{25^{47}}\times \left(\frac{6}{25}\right)^{13}\) - step10: Multiply the terms: \(\frac{206674537104\times 19^{47}}{25^{46}}\times \left(\frac{6}{25}\right)^{13}\) - step11: Evaluate the power: \(\frac{206674537104\times 19^{47}}{25^{46}}\times \frac{6^{13}}{25^{13}}\) - step12: Multiply the fractions: \(\frac{206674537104\times 19^{47}\times 6^{13}}{25^{46}\times 25^{13}}\) - step13: Multiply: \(\frac{206674537104\times 19^{47}\times 6^{13}}{25^{59}}\) The probability that exactly 47 small business owners out of 60 do not have a college degree is approximately 0.1133.