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GATE 2025 CE (CE2) – Question 22

Engineering Mathematics · Probability and Statistics: Probability, conditional probability, descriptive statistics · 1 mark · Multiple select

Given that A and B are not null sets, which of the following statements regarding probability is/are CORRECT?

  1. $P(A \cap B) = P(A)P(B)$, if A and B are mutually exclusive.
  2. Conditional probability, $P(A|B) = 1$ if $B \subset A$.
  3. $P(A \cup B) = P(A) + P(B)$, if A and B are mutually exclusive.
  4. $P(A \cap B) = 0$, if A and B are independent.

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Show answer and explanation

Correct answer: (B) Conditional probability, $P(A|B) = 1$ if $B \subset A$.; (C) $P(A \cup B) = P(A) + P(B)$, if A and B are mutually exclusive.

Explanation

For mutually exclusive events $P(A \cap B) = 0$, which is not $P(A)P(B)$ when both events are possible, so A is false. If $B \subset A$ then $A \cap B = B$ and $P(A|B) = \frac{P(B)}{P(B)} = 1$, so B is true. The addition rule for mutually exclusive events holds, so C is true. For independent events with non-zero probabilities $P(A \cap B) = P(A)P(B) > 0$, so D is false.