This problem deals with binomial probability, where we are asked to find the probability that at least 4 out of 9 human resource managers say that job applicants should follow up within two weeks.

Given:

• The probability of success $p = 0.46$ (where success is defined as a manager saying applicants should follow up within two weeks).
• The number of trials $n = 9$ (since 9 managers are selected).
• We are looking for the probability that at least 4 managers agree, i.e., $P(X \geq 4)$.

We can calculate this by first finding the complementary probability for $X < 4$, then subtracting that from 1:

$P(X \geq 4) = 1 - P(X < 4)$ $P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

For each value of $X$, the binomial probability formula is:

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

where $\binom{n}{k}$ is the binomial coefficient.

Let's compute the individual probabilities and then subtract their sum from 1.

1. $P(X = 0) = \binom{9}{0} \cdot (0.46)^0 \cdot (0.54)^9$
2. $P(X = 1) = \binom{9}{1} \cdot (0.46)^1 \cdot (0.54)^8$
3. $P(X = 2) = \binom{9}{2} \cdot (0.46)^2 \cdot (0.54)^7$
4. $P(X = 3) = \binom{9}{3} \cdot (0.46)^3 \cdot (0.54)^6$

Summing these probabilities and subtracting from 1 gives the probability of at least 4 managers agreeing.

Let me compute that for you.The probability that at least 4 out of 9 human resource managers say that job applicants should follow up within two weeks is approximately 0.6614 (rounded to four decimal places).

Would you like a more detailed breakdown of the calculations, or do you have any further questions?

Here are some related questions to consider:

1. What is the probability that exactly 5 managers say applicants should follow up within two weeks?
2. How does the probability change if 12 managers are randomly selected instead of 9?
3. What is the expected number of managers who believe applicants should follow up within two weeks?
4. What is the probability that fewer than 2 managers believe applicants should follow up within two weeks?
5. How would the outcome change if the probability of success was increased to 50%?

Tip: The binomial distribution is ideal for scenarios where you have a fixed number of independent trials with only two possible outcomes (like success or failure).