You are looking for a great person to spend your life with T

You are looking for a great person to spend your life with. The prior probability that someone is a great match is 0.1 (and I’m being generous here). If the person is a great match, their probability of having a good job is 0.8, whereas 30% of people who are not a great match also have a good job. On the first date you learn that the person has a good job. What is the probability that they are a great match? A. 0.2 B. 0.34 C. 0 D. 0.1 E. 0.23

Solution

Let

J = good job
M = good match

Thus,

P(M|J) = P(M) P(J|M) / P(J)

and as

P(J) = P(M) P(J|M) + P(M\') P(J|M\') = 0.1*0.8 + (1-0.1)*0.30 = 0.35

Thus,

P(M|J) = P(M) P(J|M) / P(J)

= 0.1*0.8/0.35

= 0.228571429 = 0.23 [ANSWER, OPTION E]