GATE CS 2017 Set 2 Q26 — Conditional Probability and Bayes Theorem

Let $P$ and $Q$ denote the events that $P$ and $Q$ apply for the job, respectively. Let $P^c$ and $Q^c$ denote the complementary events (not applying for the job).

We are given the following probabilities:

• $P(P) = \dfrac{1}{4}$
• $P(P \mid Q) = \dfrac{1}{2}$
• $P(Q \mid P) = \dfrac{1}{3}$

We need to find the conditional probability $P(P^c \mid Q^c)$.

Step 1: Find the joint probability $P(P \cap Q)$

Using the multiplication rule of probability:

Step 2: Find the probability $P(Q)$

Using the definition of conditional probability for $P(P \mid Q)$:

Step 3: Find the union probability $P(P \cup Q)$

Step 4: Find the required probability $P(P^c \mid Q^c)$

By De Morgan's laws, the joint probability of neither event occurring is:

The probability of $Q$ not applying is:

Now, calculate the conditional probability:

Thus, the correct option is A.