Let's use Bayes' theorem to find the probability that a person is qualified given that he or she was approved by the manager:
P(qualified | approved) = P(approved | qualified) * P(qualified) / P(approved)
We are given:
P(qualified) = 0.414 (the probability that a person is qualified)
P(approved | qualified) = 0.226 (the probability that the manager approves a qualified person)
P(approved | unqualified) = 0.298 (the probability that the manager approves an unqualified person)
To find P(approved), we can use the law of total probability:
P(approved) = P(approved | qualified) * P(qualified) + P(approved | unqualified) * P(unqualified)
We know that the complement of "qualified" is "unqualified", so:
P(unqualified) = 1 - P(qualified) = 1 - 0.414 = 0.586
Plugging in the given values, we get:
P(approved) = 0.226 * 0.414 + 0.298 * 0.586 = 0.276908
Now we can use Bayes' theorem to find the probability that a person is qualified given that he or she was approved by the manager:
P(qualified | approved) = 0.226 * 0.414 / 0.276908 ≈ 0.338
Therefore, the probability that a person is qualified given that he or she was approved by the manager is approximately 0.338.