This problem follows a binomial probability distribution, where:

• $n = 13$ (total number of trials, i.e., HR managers surveyed)
• $p = 0.58$ (probability of success, i.e., an HR manager saying applicants should follow up)
• We need to find $P(X < 3)$, i.e., the probability that fewer than 3 HR managers say applicants should follow up.

Step 1: Define the Binomial Probability Formula

The probability mass function (PMF) of a binomial distribution is:

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

where $\binom{n}{k}$ is the combination formula:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Step 2: Compute $P(X < 3)$

This means calculating:

$P(X = 0) + P(X = 1) + P(X = 2)$

We will now compute these values.

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Let's calculate using Python: math import comb

Given values

n = 13 # Total trials p = 0.58 # Probability of success q = 1 - p # Probability of failure

Compute probabilities for X = 0, 1, and 2

P_X_0 = comb(n, 0) * (p ** 0) * (q ** (n - 0)) P_X_1 = comb(n, 1) * (p ** 1) * (q ** (n - 1)) P_X_2 = comb(n, 2) * (p ** 2) * (q ** (n - 2))

Total probability for P(X < 3)

P_less_than_3 = P_X_0 + P_X_1 + P_X_2 P_less_than_3The probability that fewer than 3 HR managers say applicants should follow up within two weeks is 0.0021 (rounded to four decimal places).

Would you like a step-by-step breakdown of the calculations?

Related Questions:

1. How do we approximate binomial probabilities using the normal distribution?
2. What is the expected number of HR managers who believe in following up within two weeks?
3. How does changing the sample size affect the probability calculation?
4. What is the probability that at least 5 HR managers say applicants should follow up?
5. How does the probability change if the success rate drops to 50%?

Tip:

For large sample sizes ($n > 30$), the normal approximation to the binomial distribution is often used to simplify calculations.