P and Q are considering to apply for job. The probability that p applies for job is 1/4. The probability that P applies for job given that Q applies for the job 1/2 and The probability that Q applies for job given that P applies for the job 1/3.The probability that P does not apply for job given that Q does not apply for the job .a)4/5b)5/6c)7/8d)11/12Correct answer is option 'A'. Can you explain this answer?

Question

Answer

To solve this problem, we can use the concepts of conditional probability and the multiplication rule.

Let's define the events:
P: P applies for the job
Q: Q applies for the job

We are given the following probabilities:
P(P) = 1/4 (probability that P applies for the job)
P(P|Q) = 1/2 (probability that P applies for the job given that Q applies for the job)
P(Q|P) = 1/3 (probability that Q applies for the job given that P applies for the job)

We need to find the probability that P does not apply for the job given that Q does not apply for the job, which can be written as P(not P|not Q).

We can use the formula for conditional probability to find P(P and Q):
P(P and Q) = P(Q) * P(P|Q) = P(P) * P(Q|P)

Now, let's calculate P(P and Q):
P(P and Q) = (1/4) * (1/3) = 1/12

1. P(P and Q) = P(Q) * P(P|Q)
1/12 = P(not Q) * (1/2)
P(not Q) = (1/12) * (2) = 1/6

2. P(P and Q) = P(P) * P(Q|P)
1/12 = (1/4) * (1/3)
P(not P) = (1/12) * (3) = 1/4

Now, we can calculate P(not P|not Q) using the formula for conditional probability:
P(not P|not Q) = P(not P and not Q) / P(not Q)

P(not P and not Q) = P(not P) * P(not Q) = (1/4) * (1/6) = 1/24

P(not P|not Q) = (1/24) / (1/6) = 1/24 * 6/1 = 1/4

Therefore, the probability that P does not apply for the job given that Q does not apply for the job is 1/4.

The correct answer is option A) 4/5, which might be a mistake in the given options. The calculated probability is 1/4, not 4/5.

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