GATE CS 2014 Set 3 Q.58 (CS Q.48): Probability and Statistics

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 _____.

Your answer

Official answer and solution

Official answer: 0.25
From: Official GATE 2014 answer key

Solution

Use the axioms of probability to express P(B) in terms of P(A) and maximize the resulting quadratic function.

Events $A$ and $B$ are mutually exclusive, so $A \cap B = \emptyset$ .
Their union is the entire sample space: $A \cup B = S$ .
By the addition rule for mutually exclusive events:
$P(A \cup B) = P(A) + P(B) = P(S) = 1.$
Let $p = P(A)$ , then $P(B) = 1 - p$ , where $0 \le p \le 1$ .
Maximize the Product:
$f(p) = P(A)P(B) = p(1 - p) = p - p^2.$
Differentiating with respect to $p$ :
$f'(p) = 1 - 2p = 0 \implies p = \frac{1}{2} = 0.5.$
Second derivative: $f''(p) = -2 < 0$ , confirming a local maximum.
The maximum value is:
$f(0.5) = 0.5 \times 0.5 = 0.25.$

Final answer

0.25

Common trap

Thinking P(A) and P(B) are independent rather than mutually exclusive.

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

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 3
Question: 58 of 65, printed as CS Q.48
Section: Computer Science and Information Technology

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