GATE CS 2008 Q27 — Conditional Probability and Bayes Theorem

Let $C$ denote studying Computer Science and $M$ denote studying Mathematics.

From the problem statement, we have the following transition probabilities:

• If she studies $C$ on a day:
  • Probability of studying $M$ the next day: $P(M \mid C) = 0.6$
  • Probability of studying $C$ the next day: $P(C \mid C) = 1 - 0.6 = 0.4$
• If she studies $M$ on a day:
  • Probability of studying $C$ the next day: $P(C \mid M) = 0.4$
  • Probability of studying $M$ the next day: $P(M \mid M) = 1 - 0.4 = 0.6$

We are given that Aishwarya studies Computer Science on Monday ($X_{\text{Mon}} = C$). We want to find the probability that she studies Computer Science on Wednesday ($X_{\text{Wed}} = C$).

There are two mutually exclusive paths from Monday to Wednesday:

1. Path 1: Monday ($C$) $\to$ Tuesday ($C$) $\to$ Wednesday ($C$) $$P(C \to C \to C) = P(C \mid C) \times P(C \mid C) = 0.4 \times 0.4 = 0.16$$
2. Path 2: Monday ($C$) $\to$ Tuesday ($M$) $\to$ Wednesday ($C$) $$P(C \to M \to C) = P(M \mid C) \times P(C \mid M) = 0.6 \times 0.4 = 0.24$$

Adding the probabilities of these two disjoint paths:

Alternative Elegant Method (Markov Chain Transition):

The one-step transition probability matrix $P$ (with states $C$ and $M$) is:

Since both rows of the transition matrix are identical, the probability of transitioning to state $C$ on any subsequent day is independent of the current state and is always $0.4$.