Josh question on Conditional Probability

In ABC firm, prior experience suggest that 60% of all applicants for management trainee positions will be successful. Furthermore, 85% of applicants who turn out to be successful trainees pass the special aptitude test, while 70% of those who fail the special aptitude test also fail as trainees.

a- What is the probability that an applicant for a management trainee position will pass the special aptitude test?

Let S be the event that an applicant is successful after training.

Let T be the event that an applicant passed the aptitude test

The Venn looks like this:

Step 1: find P(S T ) = P(S) × P(T | S) = .60 × .85 = .51

Step 2: since P(S) = .60, P(S T ^c) = .60 – .51 = .09

Step 3: P(S ^c | T ^c ) = .70, P(S | T ^c ) = 1 – .70 = .30
Step 4: since P(S T ^c) = .09, P(S | T ^c) = .30
and P(T ^c ) =

This means that the probability of failing the test is 0.30
so the probability of passing the test is 1 – .30 = .70

The probability of passing the aptitude test is 0.70.

b- If an applicant for a management trainee position passes the aptitude test, what is the probability that he/she will be a successful trainee?

solution: We want P(S | T) = P(S T ) / P(T) = .51 ÷ .70 = 0.7286

c - Based on these probabilities, do you think that the special aptitude test is a valuable way to screen applicants for the management trainee positions? Explain numerically.

solution: Since the probability of success for applicants who pass the test is nearly 73%, compared to 60% -- the probability of success for all applicants, the test is a valuable way to screen applicants.