\( 1 \leftarrow \) Assume that when human resource managers are randomly selected, \( 49 \% \) say job applicants should follow up within two weeks. If 14 human resource managers are randomly selected, find the probability that fewer than 3 of them say job applicants should follow up within two weeks. The probability is \( \square \) (Round to four decimal places as needed.)

Given that 49% of human resource managers say job applicants should follow up within two weeks, we can assume that the probability of a human resource manager saying job applicants should follow up within two weeks is 0.49. We are asked to find the probability that fewer than 3 out of 14 randomly selected human resource managers say job applicants should follow up within two weeks. Let's denote the probability of a human resource manager saying job applicants should follow up within two weeks as \( p = 0.49 \). The probability that fewer than 3 out of 14 human resource managers say job applicants should follow up within two weeks can be calculated using the binomial distribution formula: \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \] where \( X \) is the number of human resource managers who say job applicants should follow up within two weeks. Using the binomial distribution formula, we can calculate the probabilities as follows: \[ P(X = 0) = \binom{14}{0} \times (0.49)^0 \times (1-0.49)^{14-0} \] \[ P(X = 1) = \binom{14}{1} \times (0.49)^1 \times (1-0.49)^{14-1} \] \[ P(X = 2) = \binom{14}{2} \times (0.49)^2 \times (1-0.49)^{14-2} \] Now, we can calculate the probabilities and sum them up to find the probability that fewer than 3 out of 14 human resource managers say job applicants should follow up within two weeks. Calculate the value by following steps: - step0: Calculate: \( { }_{14}C_{0}\times 0.49^{0}\left(1-0.49\right)^{14-0}\) - step1: Evaluate the power: \( { }_{14}C_{0}\times 1\times \left(1-0.49\right)^{14-0}\) - step2: Subtract the numbers: \( { }_{14}C_{0}\times 1\times 0.51^{14-0}\) - step3: Remove 0: \( { }_{14}C_{0}\times 1\times 0.51^{14}\) - step4: Expand the expression: \(\frac{14!}{0!\times \left(14-0\right)!}\times 1\times 0.51^{14}\) - step5: Remove 0: \(\frac{14!}{0!\times 14!}\times 1\times 0.51^{14}\) - step6: Reduce the fraction: \(\frac{14!}{14!}\times 1\times 0.51^{14}\) - step7: Divide the terms: \(1\times 1\times 0.51^{14}\) - step8: Convert the expressions: \(1\times 1\times \left(\frac{51}{100}\right)^{14}\) - step9: Multiply the terms: \(\left(\frac{51}{100}\right)^{14}\) - step10: Use the properties of exponents: \(\frac{51^{14}}{100^{14}}\) Calculate or simplify the expression \( \binom{14}{1} * (0.49)^1 * (1-0.49)^{14-1} \). Calculate the value by following steps: - step0: Calculate: \( { }_{14}C_{1}\times 0.49^{1}\left(1-0.49\right)^{14-1}\) - step1: Subtract the numbers: \( { }_{14}C_{1}\times 0.49^{1}\times 0.51^{14-1}\) - step2: Subtract the numbers: \( { }_{14}C_{1}\times 0.49^{1}\times 0.51^{13}\) - step3: Expand the expression: \(\frac{14!}{1!\times \left(14-1\right)!}\times 0.49^{1}\times 0.51^{13}\) - step4: Subtract the numbers: \(\frac{14!}{1!\times 13!}\times 0.49^{1}\times 0.51^{13}\) - step5: Reduce the fraction: \(\frac{14!}{13!}\times 0.49^{1}\times 0.51^{13}\) - step6: Reduce the fraction: \(14\times 0.49^{1}\times 0.51^{13}\) - step7: Calculate: \(14\times \frac{49}{100}\times 0.51^{13}\) - step8: Convert the expressions: \(14\times \frac{49}{100}\left(\frac{51}{100}\right)^{13}\) - step9: Multiply the terms: \(\frac{343}{50}\left(\frac{51}{100}\right)^{13}\) - step10: Evaluate the power: \(\frac{343}{50}\times \frac{51^{13}}{100^{13}}\) - step11: Multiply the fractions: \(\frac{343\times 51^{13}}{50\times 100^{13}}\) Calculate or simplify the expression \( \binom{14}{2} * (0.49)^2 * (1-0.49)^{14-2} \). Calculate the value by following steps: - step0: Calculate: \( { }_{14}C_{2}\times 0.49^{2}\left(1-0.49\right)^{14-2}\) - step1: Subtract the numbers: \( { }_{14}C_{2}\times 0.49^{2}\times 0.51^{14-2}\) - step2: Subtract the numbers: \( { }_{14}C_{2}\times 0.49^{2}\times 0.51^{12}\) - step3: Expand the expression: \(\frac{14!}{2!\times \left(14-2\right)!}\times 0.49^{2}\times 0.51^{12}\) - step4: Subtract the numbers: \(\frac{14!}{2!\times 12!}\times 0.49^{2}\times 0.51^{12}\) - step5: Reduce the fraction: \(\frac{14\times 13}{2!}\times 0.49^{2}\times 0.51^{12}\) - step6: Multiply the numbers: \(\frac{182}{2!}\times 0.49^{2}\times 0.51^{12}\) - step7: Divide the terms: \(91\times 0.49^{2}\times 0.51^{12}\) - step8: Convert the expressions: \(91\left(\frac{49}{100}\right)^{2}\times 0.51^{12}\) - step9: Convert the expressions: \(91\left(\frac{49}{100}\right)^{2}\left(\frac{51}{100}\right)^{12}\) - step10: Multiply the terms: \(\frac{218491}{10000}\left(\frac{51}{100}\right)^{12}\) - step11: Evaluate the power: \(\frac{218491}{10000}\times \frac{51^{12}}{100^{12}}\) - step12: Multiply the fractions: \(\frac{218491\times 51^{12}}{10000\times 100^{12}}\) - step13: Multiply: \(\frac{218491\times 51^{12}}{100^{14}}\) Calculate or simplify the expression \( 8.053459\times 10^{-5} + 0.001083 + 0.006765 \). Calculate the value by following steps: - step0: Calculate: \(8.053459\times 10^{-5}+0.001083+0.006765\) - step1: Add the numbers: \(8.053459\times 10^{-5}+0.007848\) - step2: Evaluate: \(\frac{8053459}{10^{11}}+0.007848\) - step3: Convert the expressions: \(\frac{8053459}{10^{11}}+\frac{981}{125000}\) - step4: Reduce fractions to a common denominator: \(\frac{8053459}{10^{11}}+\frac{981\times 800000}{125000\times 800000}\) - step5: Calculate: \(\frac{8053459}{50^{3}\times 800000}+\frac{784800000}{50^{3}\times 800000}\) - step6: Transform the expression: \(\frac{8053459+784800000}{50^{3}\times 800000}\) - step7: Add the numbers: \(\frac{792853459}{50^{3}\times 800000}\) The probability that fewer than 3 out of 14 human resource managers say job applicants should follow up within two weeks is approximately 0.007929.