Assume that when human resource managers are randomly​ selected, 41​% say job applicants should follow up within two weeks. If 25 human resource managers are randomly​ selected, find the probability that exactly 17 of them say job applicants should follow up within two weeks.
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Part 1
The probability is 
  
enter your response here.
​(Round to four decimal places as​ needed.)

This problem follows a binomial probability distribution, where:

- \( n = 25 \) (the number of trials, i.e., the number of human resource managers),
- \( p = 0.41 \) (the probability that a manager says job applicants should follow up within two weeks),
- \( x = 17 \) (the number of successes we are interested in),
- \( q = 1 - p = 0.59 \) (the probability of failure).

The binomial probability formula is:

\[
P(X = x) = \binom{n}{x} p^x q^{n-x}
\]

Where \( \binom{n}{x} \) is the binomial coefficient:

\[
\binom{n}{x} = \frac{n!}{x!(n-x)!}
\]

For this case, the probability of exactly 17 managers saying job applicants should follow up within two weeks is:

\[
P(X = 17) = \binom{25}{17} (0.41)^{17} (0.59)^{8}
\]

We will calculate this step by step.

1. Calculate the binomial coefficient:
\[
\binom{25}{17} = \frac{25!}{17!(25-17)!} = \frac{25!}{17!8!}
\]

2. Calculate \( p^{17} = (0.41)^{17} \) and \( q^{8} = (0.59)^{8} \).

3. Multiply all terms together to get the final probability.

Let me compute that for you:The probability that exactly 17 out of 25 human resource managers say job applicants should follow up within two weeks is \( 0.0042 \), rounded to four decimal places.

Would you like further details or have any questions?

Here are 5 related questions you could explore:
1. What is the probability that fewer than 17 managers say applicants should follow up within two weeks?
2. How do you calculate the expected value in a binomial distribution?
3. What is the variance of this binomial distribution?
4. What would the probability be for exactly 10 managers saying applicants should follow up within two weeks?
5. How does the shape of the binomial distribution change as the probability \( p \) varies?

**Tip:** For large sample sizes, binomial distributions can be approximated using the normal distribution!

Assume that when human resource managers are randomly selected, 41% say job applicants should follow up within two weeks. If 25 human resource managers are randomly selected, find the probability that exactly 17 of them say job applicants should follow up within two weeks.

Calculate the probability that exactly 17 out of 25 randomly selected human resource managers say job applicants should follow up within two weeks, using binomial distribution. This example covers probability concepts and binomial coefficients, ideal for Grades 10-12 students.

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