GATE CS 2014 Set 3 Q48 — Probability: Random Variables and Distributions

Given that $A$ and $B$ are mutually exclusive events and $A \cup B = S$, we have:

Since $A$ and $B$ are mutually exclusive ($A \cap B = \emptyset$), the probability of their union is the sum of their individual probabilities:

Let $P(A) = x$. Since probabilities must lie between $0$ and $1$, we have $0 \le x \le 1$. Then, $P(B) = 1 - x$.

We want to maximize the product $P(A)P(B)$:

To find the value of $x$ that maximizes $f(x)$, we take the first derivative with respect to $x$ and set it to zero:

Since the second derivative $f''(x) = -2 < 0$, $x = 0.5$ indeed yields the maximum value.

Substituting $x = 0.5$ back into the product: