\( 1 \leftarrow \) Assume that when human resource managers are randomly selected, \( 52 \% \) say job applicants should follow up within two weeks. If 10 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 (Round to four decimal places as needed.)

To solve this problem, we can use the binomial probability formula. The scenario describes a binomial distribution where: - \( n = 10 \) (the number of trials, or human resource managers selected) - \( p = 0.52 \) (the probability of success, which is the probability that a manager says job applicants should follow up within two weeks) - \( k \) (the number of successes, which in this case is the number of managers who say applicants should follow up) We want to find the probability that fewer than 3 managers say job applicants should follow up within two weeks, which means we need to calculate \( P(X < 3) \). This can be expressed as: \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \] The binomial probability formula is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient, calculated as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Now, we will calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \). ### Step 1: Calculate \( P(X = 0) \) \[ P(X = 0) = \binom{10}{0} (0.52)^0 (0.48)^{10} = 1 \cdot 1 \cdot (0.48)^{10} \] Calculating \( (0.48)^{10} \): \[ (0.48)^{10} \approx 0.006044 \] So, \[ P(X = 0) \approx 0.006044 \] ### Step 2: Calculate \( P(X = 1) \) \[ P(X = 1) = \binom{10}{1} (0.52)^1 (0.48)^{9} = 10 \cdot 0.52 \cdot (0.48)^{9} \] Calculating \( (0.48)^{9} \): \[ (0.48)^{9} \approx 0.012605 \] So, \[ P(X = 1) \approx 10 \cdot 0.52 \cdot 0.012605 \approx 0.0656 \] ### Step 3: Calculate \( P(X = 2) \) \[ P(X = 2) = \binom{10}{2} (0.52)^2 (0.48)^{8} = 45 \cdot (0.52)^2 \cdot (0.48)^{8} \] Calculating \( (0.48)^{8} \): \[ (0.48)^{8} \approx 0.026315 \] So, \[ P(X = 2) \approx 45 \cdot (0.52)^2 \cdot 0.026315 \] Calculating \( (0.52)^2 \): \[ (0.52)^2 = 0.2704 \] Now, \[ P(X = 2) \approx 45 \cdot 0.2704 \cdot 0.026315 \approx 0.3241 \] ### Step 4: Combine the probabilities Now we can sum these probabilities: \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \] \[ P(X < 3) \approx 0.006044 + 0.0656 + 0.3241 \approx 0.3957 \] ### Final Answer Thus, the probability that fewer than 3 of the selected human resource managers say job applicants should follow up within two weeks is approximately: \[ \boxed{0.3957} \]