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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

Options

  1. A4 5
  2. B7 8
  3. C11 12
  4. D5 6

Correct answer

A. 4 5

Step-by-step solution

Let P be the event that P applies for the job and Q be the event that Q applies for the job. Given probabilities are P(P) = 1 4 , P(P|Q) = 1 2 , and P(Q|P) = 1 3 . Using the definition of conditional probability, P(P Q) = P(Q|P) P(P) = 1 3 1 4 = 1 12 . Also, P(P Q) = P(P|Q) P(Q) , so 1 12 = 1 2 P(Q) , which gives P(Q) = 2 12 = 1 6 . We need to find P(P^c | Q^c) , where P^c and Q^c are the complements of events P and Q . By definition, P(P^c | Q^c) = P(P^c Q^c) P(Q^c) . Using De Morgan's Law, P(P^c Q^c) = P((P Q)^c)

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