Assume that when human resource managers are randomly selected, 61% say job applicants should follow up within two weeks. If 20 human resource managers are randomly selected, find the probability that exactly 14 of them say job applicants should follow up within two weeks.

The probability is
(Round to four decimal places as needed.)



Answer :

Sure! Let's solve this problem step-by-step.

### Step 1: Understanding the Problem

We have the following parameters:
- The probability of success, \( p = 0.61 \), which means there is a 61% chance that a human resource manager says job applicants should follow up within two weeks.
- The number of trials, \( n = 20 \), which indicates that 20 human resource managers are randomly selected.
- We are interested in the case where exactly \( k = 14 \) managers say that applicants should follow up within two weeks.

### Step 2: Binomial Distribution

We use the binomial distribution to solve this problem. The binomial distribution formula for finding the probability of exactly \( k \) successes in \( n \) independent trials is given by:

[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]

where:
- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient,
- \( p^k \) is the probability of having \( k \) successes,
- \( (1-p)^{n-k} \) is the probability of having \( n-k \) failures.

### Step 3: Plugging in the Values

Now let's plug in the values into the formula:

- \( n = 20 \)
- \( k = 14 \)
- \( p = 0.61 \)
- \( 1 - p = 0.39 \)

First, we calculate the binomial coefficient:

[tex]\[ \binom{20}{14} = \frac{20!}{14!(20-14)!} = \frac{20!}{14! \cdot 6!} \][/tex]

Next, we calculate \( p^k \) and \( (1 - p)^{n - k} \):

[tex]\[ p^{14} = 0.61^{14} \][/tex]
[tex]\[ (1 - p)^{6} = 0.39^{6} \][/tex]

### Step 4: Computing the Probability

Multiplying these components together gives us the probability \( P(X = 14) \).

After performing the calculations (simplified here for clarity), we find:

[tex]\[ P(X = 14) \approx 0.1347 \][/tex]

### Step 5: Conclusion

Thus, the probability that exactly 14 out of 20 randomly selected human resource managers say job applicants should follow up within two weeks is approximately \( 0.1347 \).

### Final Answer

The probability is [tex]\( \boxed{0.1347} \)[/tex].