Suppose that when a job candidate comes to interview for a job at, say McDonald's, the probability that the candidate will want the job (A) after the interview is .68. Also, the probability that McDonald's wants the candidate (B) is .36. The probability P(A/B) is .88.
a.) Find P(A and B)
b.) Find P(B/A)
c.) Are events A and B independent? Explain.
d.) Are events A and B mutually exclusive? Explain.
e.) What would it mean to say A and B are mutually exclusive events in this exercise?
I will be glad to critique your thinking. Hopefully, you have learned something from previous answers provided to you.
a.) To find P(A and B), we can use the formula for conditional probability: P(A and B) = P(A) * P(B/A). Given that P(A) = 0.68 and P(A/B) = 0.88, we can calculate P(A and B) as follows:
P(A and B) = P(A) * P(B/A)
= 0.68 * 0.88
= 0.5984
Therefore, P(A and B) is approximately 0.5984.
b.) To find P(B/A), we can use Bayes' theorem, which states that P(B/A) = (P(A/B) * P(B)) / P(A). Given that P(A/B) = 0.88, P(B) = 0.36, and P(A) = 0.68, we can calculate P(B/A) as follows:
P(B/A) = (P(A/B) * P(B)) / P(A)
= (0.88 * 0.36) / 0.68
= 0.4659
Therefore, P(B/A) is approximately 0.4659.
c.) Events A and B are independent if and only if P(A and B) = P(A) * P(B). We can compare these probabilities to determine if A and B are independent:
P(A) * P(B) = 0.68 * 0.36 = 0.2448
P(A and B) = 0.5984
Since P(A and B) ≠ P(A) * P(B), events A and B are not independent.
d.) Events A and B are mutually exclusive if they cannot happen at the same time, meaning their intersection is empty. In this scenario, it is possible for A and B to happen simultaneously. For example, a candidate could want the job (A) and McDonald's could want the candidate (B). Therefore, events A and B are not mutually exclusive.
e.) If events A and B were mutually exclusive in this exercise, it would mean that it is not possible for a candidate to want the job (A) and for McDonald's to want the candidate (B) at the same time. In other words, if A and B were mutually exclusive, there would be no overlap between the two events - if one event occurs, the other cannot occur. However, as we have seen, it is possible for both events A and B to happen simultaneously, so they are not mutually exclusive.