Let S be a sample space and two mutually exclusive events A and B be such that A B = S. If P( ) denotes the probability of the event, the maximum value of P(A)P(B) is …

Official answer: 0.25. Explanation: Given that A and B are mutually exclusive events, we have: A B = P(A B) = 0 Also given that A B = S, so: P(A B) = P(S) = 1 By the addition rule of probability: P(A B) = P(A) + P(B) - P(A B) 1 = P(A) + P(B) - 0 P(A) + P(B) = 1 Let P(A) = p. Then P(B) = 1 - p, where 0 p 1. We wish to maximize the product: f(p) = P(A)P(B) = p(1-p) = p - p2 Taking the derivative with respect to p and setting it to zero: f'(p) = 1 - 2p = 0 p = 0.5 Since f''(p) = -2 < 0, the function achieves its maximum at p = 0.5: f(0.5) = 0.5 (1 - 0.5) = 0.25 Thus, the maximum value of P(A)P(B) is 0.25.

Engineering Mathematics · Probability and Statistics

Official IIT answer key · IIT Kharagpur · Audited Aug 2026

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GATE CSE 2014 Set 3 Q48 · NAT · 2 marks

Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.

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Key concept

Probability and Statistics

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How GATE tests Probability and Statistics

Probability and Statistics in GATE CS is consistently tested with 2 to 4 questions per year, spanning fundamental combinatorics, conditional probability, Bayes' theorem, discrete and continuous random variables, and expectation/variance properties. A notable proportion of questions require setting up a discrete probability model (urn/coin/dice experiments or bit strings) or computing expectations using linearity of expectation and integration over piece-wise uniform intervals. The question mix favors NAT and conceptual MCQ/MSQ formats with an emphasis on exact algebraic reasoning rather than heavy numerical statistical testing.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Probability and Statistics

Official IIT answer key · IIT Kharagpur · Audited Aug 2026

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GATE CSE 2014 Set 3 Q48 · NAT · 2 marks

Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.

Numerical answer

Enter a value to check

Key concept

Probability and Statistics

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Probability and Statistics

Full Probability and Statistics guide →

More on Probability and Statistics

All 46 questions on Probability and Statistics →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.

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Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.

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Let SSS be a sample space and two mutually exclusive events AAA and BBB be such that A∪B=SA \cup B = SA∪B=S. If P(⋅)P(\cdot)P(⋅) denotes the probability of the event, the maximum value of P(A)P(B)P(A)P(B)P(A)P(B) is _________.

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Let SSS be a sample space and two mutually exclusive events AAA and BBB be such that A∪B=SA \cup B = SA∪B=S. If P(⋅)P(\cdot)P(⋅) denotes the probability of the event, the maximum value of P(A)P(B)P(A)P(B)P(A)P(B) is _________.

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Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.

Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P(\cdot)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is _________.