Assume that when human resource managers are randomly selected, \( 62 \% \) say job applicants should follow up within two weeks. If 5 human resource managers are randomly selected, find the probability that exactly 3 of them say job applicants should follow up within two weeks. The probability is (Round to four decimal places as needed.)

To find the probability that exactly 3 out of 5 human resource managers say job applicants should follow up within two weeks, we can use the binomial probability formula. Given: - Probability of a human resource manager saying job applicants should follow up within two weeks: \( p = 0.62 \) - Number of human resource managers selected: \( n = 5 \) - Number of human resource managers saying job applicants should follow up within two weeks: \( k = 3 \) The binomial probability formula is: \[ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \] Substitute the given values into the formula: \[ P(X = 3) = \binom{5}{3} \times 0.62^3 \times (1-0.62)^{5-3} \] Now, we can calculate the probability using the binomial probability formula. Calculate the value by following steps: - step0: Calculate: \( { }_{5}C_{3}\times 0.62^{3}\left(1-0.62\right)^{5-3}\) - step1: Subtract the numbers: \( { }_{5}C_{3}\times 0.62^{3}\times 0.38^{5-3}\) - step2: Subtract the numbers: \( { }_{5}C_{3}\times 0.62^{3}\times 0.38^{2}\) - step3: Expand the expression: \(\frac{5!}{3!\times \left(5-3\right)!}\times 0.62^{3}\times 0.38^{2}\) - step4: Subtract the numbers: \(\frac{5!}{3!\times 2!}\times 0.62^{3}\times 0.38^{2}\) - step5: Reduce the fraction: \(\frac{5\times 4}{2!}\times 0.62^{3}\times 0.38^{2}\) - step6: Multiply the numbers: \(\frac{20}{2!}\times 0.62^{3}\times 0.38^{2}\) - step7: Divide the terms: \(10\times 0.62^{3}\times 0.38^{2}\) - step8: Convert the expressions: \(10\left(\frac{31}{50}\right)^{3}\times 0.38^{2}\) - step9: Convert the expressions: \(10\left(\frac{31}{50}\right)^{3}\left(\frac{19}{50}\right)^{2}\) - step10: Simplify: \(10\times \frac{31^{3}}{50^{3}}\times \left(\frac{19}{50}\right)^{2}\) - step11: Multiply the terms: \(\frac{31^{3}}{12500}\left(\frac{19}{50}\right)^{2}\) - step12: Evaluate the power: \(\frac{31^{3}}{12500}\times \frac{19^{2}}{50^{2}}\) - step13: Multiply the fractions: \(\frac{31^{3}\times 19^{2}}{12500\times 50^{2}}\) - step14: Multiply: \(\frac{10754551}{31250000}\) The probability that exactly 3 out of 5 human resource managers say job applicants should follow up within two weeks is approximately 0.3441 (rounded to four decimal places).