GATE CS 2024 Set 1 Q17 — Probability: Random Variables and Distributions

We are given the following probabilities for events $A$ and $B$:

• $P(A) = 0.3$
• $P(B) = 0.5$
• $P(A \cap B) = 0.1$

Let's evaluate each option step-by-step:

1. The two events $A$ and $B$ are independent:

   Two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A) \cdot P(B)$.

   $$P(A) \cdot P(B) = 0.3 \times 0.5 = 0.15$$

   Since $P(A \cap B) = 0.1 \neq 0.15$, the events $A$ and $B$ are not independent. Thus, statement A is FALSE.

2. $P(A \cup B) = 0.7$:

   Using the addition rule of probability:

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

   $$P(A \cup B) = 0.3 + 0.5 - 0.1 = 0.7$$

   Thus, statement B is TRUE.

3. $P(A \cap B^c) = 0.2$:

   The probability of event $A$ occurring but not $B$ is given by:

   $$P(A \cap B^c) = P(A) - P(A \cap B)$$

   $$P(A \cap B^c) = 0.3 - 0.1 = 0.2$$

   Thus, statement C is TRUE.

4. $P(A^c \cap B^c) = 0.4$:

   Using De Morgan's Law:

   $$P(A^c \cap B^c) = P((A \cup B)^c) = 1 - P(A \cup B)$$

   $$P(A^c \cap B^c) = 1 - 0.7 = 0.3$$

   Since $0.3 \neq 0.4$, statement D is FALSE.