GATE CS 1999 Q2.1 — Probability: Random Variables and Distributions

We are given the following probabilities for two events $E_1$ and $E_2$:

• $P(E_1) = \frac{1}{2}$
• $P(E_2) = \frac{1}{3}$
• $P(E_1 \cap E_2) = \frac{1}{5}$

Let's evaluate the options to find the correct statement:

1. Option A: $P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2) = \frac{1}{2} + \frac{1}{3} - \frac{1}{5} = \frac{15 + 10 - 6}{30} = \frac{19}{30} \neq \frac{2}{3}$. Thus, Option A is false.
2. Options B and C: Two events $E_1$ and $E_2$ are independent if and only if $P(E_1 \cap E_2) = P(E_1) \cdot P(E_2)$. Here, $P(E_1) \cdot P(E_2) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$. Since $P(E_1 \cap E_2) = \frac{1}{5} \neq \frac{1}{6}$, the events $E_1$ and $E_2$ are not independent. Thus, Option B is false and Option C is true.
3. Option D: $P(E_1 \mid E_2) = \frac{P(E_1 \cap E_2)}{P(E_2)} = \frac{1/5}{1/3} = \frac{3}{5} \neq \frac{4}{5}$. Thus, Option D is false.