GATE CS 2024 Set 1 Q.27: Probability and Statistics

Let $A$ and $B$ be two events in a probability space with $P(A) = 0.3, P(B) = 0.5$ , and $P(A \cap B) = 0.1$ . Which of the following statements is/are TRUE?

The two events $A$ and $B$ are independent

$P(A \cup B) = 0.7$

$P(A \cap B^c) = 0.2$ , where $B^c$ is the complement of the event $B$

$P(A^c \cap B^c) = 0.4$ , where $A^c$ and $B^c$ are the complements of the events $A$ and $B$ , respectively

Official answer and solution

Official answer: B and C
From: Official GATE 2024 answer key

Solution

Check independence via $P(A)P(B)$ , union via inclusion-exclusion, and complements via set difference and De Morgan's law.

(A) is FALSE: $P(A)P(B) = 0.3 \times 0.5 = 0.15 \neq 0.1 = P(A \cap B)$ , so $A$ and $B$ are not independent.
(B) is TRUE: By inclusion-exclusion, $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.5 - 0.1 = 0.7$ .
(C) is TRUE: $P(A \cap B^c) = P(A) - P(A \cap B) = 0.3 - 0.1 = 0.2$ .
(D) is FALSE: By De Morgan's law, $P(A^c \cap B^c) = 1 - P(A \cup B) = 1 - 0.7 = 0.3 \neq 0.4$ .

Final answer

Options (B), (C)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 27 of 65
Section: Computer Science and Information Technology

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