P and Q are considering to apply for a job. The probability that P applies for the job is 1/4, the probability that P applies for the job given that Q applies for the job is 1/2, and the probability that Q applies for the job given that P applies for the job is 1/3. Then the probability that P does not apply for the job given that Q does not apply for the job is: A. 4/5 B. 1/3 C. 5/6 D. 11/12

GATE 2017 · Engineering Mathematics · Conditional Probability · medium

Answer: 4/5

1. Find P(P and Q): P(P and Q) = P(Q|P) * P(P) = (1/3) * (1/4) = 1/12.
2. Find P(Q): P(Q) = (1/12) / (1/2) = 2/12 = 1/6.
3. Find P(P^c and Q^c) via inclusion-exclusion: P(P^c and Q^c) = 1 - 1/4 - 1/6 + 1/12 = 12/12 - 3/12 - 2/12 + 1/12 = 8/12 = 2/3.
4. Compute P(P^c | Q^c): P(Q^c) = 1 - P(Q) = 1 - 1/6 = 5/6. P(P^c | Q^c) = (2/3) / (5/6) = (2/3) * (6/5) = 12/15 = 4/5.