This is a binomial probability problem where:

• $p = 0.57$ is the probability that a randomly selected HR manager says applicants should follow up within two weeks.
• $n = 10$ is the number of HR managers randomly selected.
• We are asked to find the probability that fewer than 3 of them, i.e., 0, 1, or 2, say applicants should follow up within two weeks.

Binomial Distribution Formula

The probability of exactly $k$ successes in $n$ trials is given by the binomial probability formula:

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

Where:

• $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ is the binomial coefficient,
• $p$ is the probability of success (in this case, 0.57),
• $n$ is the number of trials (10 in this case),
• $k$ is the number of successes (0, 1, or 2 in this case).

Step-by-Step Calculation

We need to calculate the probability for $X < 3$, so we will calculate:

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

1. Calculate $P(X = 0)$

$P(X = 0) = \binom{10}{0} (0.57)^0 (0.43)^{10} = 1 \times 1 \times (0.43)^{10}$

2. Calculate $P(X = 1)$

$P(X = 1) = \binom{10}{1} (0.57)^1 (0.43)^9 = 10 \times 0.57 \times (0.43)^9$

3. Calculate $P(X = 2)$

$P(X = 2) = \binom{10}{2} (0.57)^2 (0.43)^8 = 45 \times (0.57)^2 \times (0.43)^8$

We will now perform these calculations.

Let me calculate these for you. math

Given values

p = 0.57 q = 1 - p n = 10

Calculate P(X = 0), P(X = 1), and P(X = 2)

P_X_0 = math.comb(n, 0) * (p0) * (q10) P_X_1 = math.comb(n, 1) * (p1) * (q9) P_X_2 = math.comb(n, 2) * (p2) * (q8)

Total probability P(X < 3)

P_X_less_than_3 = P_X_0 + P_X_1 + P_X_2

P_X_less_than_3The probability that fewer than 3 of the 10 randomly selected HR managers say job applicants should follow up within two weeks is approximately 0.0202 (rounded to four decimal places).

Would you like further details or clarification on any part of the calculation?

Here are some related questions that might help you dive deeper into this topic:

1. How do you interpret the binomial probability distribution in real-world scenarios?
2. What is the meaning of "success" and "failure" in binomial probability problems?
3. How do changes in the probability $p$ affect the shape of the binomial distribution?
4. How can you use the normal approximation to the binomial distribution for large $n$?
5. What is the relationship between the binomial distribution and the Poisson distribution?

Tip: In real-world applications, understanding the binomial distribution can be helpful when making decisions that involve yes/no outcomes, such as quality control or surveys.