Question

Of all the people applying for a certain​ job, 80​%are qualified and 20​% 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 qualified if he or she was approved by the manager

The probability is..?

Answer 1

We are given here that:
P( qualified ) = 0.8,
P( not qualified ) = 0.2

Also, P( approve | qualified ) = 0.8 and P( approve | not qualified ) = 0.1

Using law of total probability, we get here:
P( approve) = P( approve | qualified )P( qualified ) + P( approve | not qualified )P( not qualified )

P( approve) = 0.8*0.8 + 0.2*0.1 = 0.64 + 0.02 = 0.66

Given that the the person is approved by manager, probability that the person is qualified is computed using Bayes theorem here as:

P( qualified | approve ) = P( approve | qualified )P( qualified ) / P( approve)

= 0.64 /0.66

= 0.9697

Therefore 0.9697 is the required probability here.