Answer:
Explanation:
To find the probability that fewer than 3 of the 14 randomly selected human resource managers say job applicants should follow up within two weeks, you can use the binomial probability formula. In this case, it's a binomial distribution because there are only two possible outcomes for each manager: they either say job applicants should follow up within two weeks (success) or they don't (failure).
The formula for the probability of \(k\) successes in \(n\) trials with a probability of success \(p\) is:
\[P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\]
Where:
- \(P(X = k)\) is the probability of getting exactly \(k\) successes.
- \(\binom{n}{k}\) is the binomial coefficient, also known as "n choose k," which represents the number of ways to choose \(k\) successes out of \(n\) trials.
- \(p\) is the probability of success for a single trial.
- \(1-p\) is the probability of failure for a single trial.
- \(n\) is the total number of trials.
In your case:
- \(n\) (the number of trials) is 14 (since you're selecting 14 HR managers).
- \(p\) (the probability of success on a single trial) is 0.52 (52% expressed as a decimal).
- \(k\) can be 0, 1, or 2 because you want fewer than 3 HR managers to say applicants should follow up within two weeks.
Now, calculate the probabilities for each of these cases and add them together:
1. For \(k = 0\):
\[P(X = 0) = \binom{14}{0} \cdot (0.52)^0 \cdot (1-0.52)^{14-0}\]
2. For \(k = 1\):
\[P(X = 1) = \binom{14}{1} \cdot (0.52)^1 \cdot (1-0.52)^{14-1}\]
3. For \(k = 2\):
\[P(X = 2) = \binom{14}{2} \cdot (0.52)^2 \cdot (1-0.52)^{14-2}\]
Now, calculate each of these probabilities, and then add them together to find the total probability that fewer than 3 HR managers say applicants should follow up within two weeks:
\[P(\text{Fewer than 3}) = P(X = 0) + P(X = 1) + P(X = 2)\]
Calculate each term using the formula, and then add them up to get the final probability.