P and Q are considering to apply for a job. The probability that P applies for…
2017
P and Q are considering to apply for a job. The probability that P applies for the job is \(\frac {1} {4}\), the probability that P applies for the job given that Q applies for the job is \(\frac {1} {2}\), and the probability that Q applies for the job given that P applies for the job is \(\frac {1} {3}\). Then the probability that P does not apply for the job given that Q does not apply for this job is
- A.
\(\frac {4} {5}\) - B.
\(\frac {5} {6}\) - C.
\(\frac {7} {8}\) - D.
\(\frac {11} {12}\)
Attempted by 53 students.
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Correct answer: A
Given: P(P) = 1/4, P(P | Q) = 1/2, P(Q | P) = 1/3. We want P(not P | not Q).
Compute the joint probability P(P ∩ Q) using P(Q | P):
P(P ∩ Q) = P(P) × P(Q | P) = 1/4 × 1/3 = 1/12.
Find P(Q) using P(P | Q):
P(P | Q) = P(P ∩ Q) / P(Q) ⇒ P(Q) = P(P ∩ Q) / P(P | Q) = (1/12) / (1/2) = 1/6.
Compute complements and remaining joint probabilities:
P(Q') = 1 − P(Q) = 5/6.
P(P ∩ Q') = P(P) − P(P ∩ Q) = 1/4 − 1/12 = 1/6.
P(not P ∩ not Q) = P(Q') − P(P ∩ Q') = 5/6 − 1/6 = 2/3.
Finally compute the required conditional probability:
P(not P | not Q) = P(not P ∩ not Q) / P(not Q) = (2/3) / (5/6) = 4/5.
Answer: 4/5