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Part 1
Assume that when human resource managers are randomly​ selected, 56​% say job applicants should follow up within two weeks. If 6 human resource managers are randomly​ selected, find the probability that exactly 2 of them say job applicants should follow up within two weeks.
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Part 1
The probability is 
  
enter your response here.

To solve this, we can use the **binomial probability formula**, which is appropriate when we have a fixed number of trials, two possible outcomes (success or failure), and constant probability of success across trials.

The binomial probability formula is:

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

Where:
- \( n \) = number of trials (in this case, 6 human resource managers),
- \( k \) = number of successes (we are looking for exactly 2 who say applicants should follow up),
- \( p \) = probability of success (56% = 0.56),
- \( \binom{n}{k} \) = "n choose k", the binomial coefficient.

### Step-by-step calculation:

1. **Number of trials** \( n = 6 \)
2. **Number of successes** \( k = 2 \)
3. **Probability of success** \( p = 0.56 \)
4. **Probability of failure** \( 1 - p = 0.44 \)

The binomial coefficient \(\binom{6}{2}\) is calculated as:

\[
\binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]

Now, plug these values into the binomial probability formula:

\[
P(X = 2) = 15 \cdot (0.56)^2 \cdot (0.44)^4
\]

Let's calculate this value.The probability that exactly 2 out of 6 human resource managers say job applicants should follow up within two weeks is approximately **0.176** or **17.6%**.

Would you like further details on this calculation or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What is the probability that exactly 3 managers say job applicants should follow up within two weeks?
2. How does changing the number of selected managers affect the probability?
3. What is the probability that none of the managers say applicants should follow up within two weeks?
4. How can we interpret the cumulative probability of 2 or fewer managers giving this response?
5. What assumptions must hold true for this binomial model to be valid?

**Tip:** In binomial distributions, if the probability of success is close to 0.5, the distribution tends to be more symmetric.

Assume that when human resource managers are randomly selected, 56% say job applicants should follow up within two weeks. If 6 human resource managers are randomly selected, find the probability that exactly 2 of them say job applicants should follow up within two weeks.

Find the probability that exactly 2 out of 6 human resource managers say job applicants should follow up within two weeks, given a 56% success rate. This problem explores key concepts in probability and binomial distribution, with a step-by-step solution suitable for Grades 10-12.

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