Question

Of all the people applying for a certain job, 75% are qualified and 25% are not. The personnel manager claims that she approves qualified people 80% of the time; she approves unqualified people 10% of the time. Find the probability that a person is not qualified if he or she was approved by the manager. The probability is (Type an integer or decimal rounded to four decimal places as needed.)

Answer 1

Answer: Therefore, the probability that a person is not qualified if they were approved by the manager is approximately 0.0625 (rounded to four decimal places).

Explanation:

Let's denote the events as follows:

Q: The person is qualified

NQ: The person is not qualified

A: The person is approved by the manager

We are given the following probabilities:

P(Q) = 0.75 (probability that a person is qualified)

P(NQ) = 0.25 (probability that a person is not qualified)

P(A|Q) = 0.80 (probability that a qualified person is approved)

P(A|NQ) = 0.10 (probability that a not qualified person is approved)

We want to find P(NQ|A), the probability that a person is not qualified given that they were approved by the manager.

By applying Bayes' theorem, we have:

P(NQ|A) = (P(A|NQ) * P(NQ)) / [P(A|Q) * P(Q) + P(A|NQ) * P(NQ)]

Substituting the given values, we get:

P(NQ|A) = (0.10 * 0.25) / [(0.80 * 0.75) + (0.10 * 0.25)]

Calculating this expression, we find:

P(NQ|A) ≈ 0.0625

Therefore, the probability that a person is not qualified if they were approved by the manager is approximately 0.0625 (rounded to four decimal places).