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 _________.

Given that $A$ and $B$ are mutually exclusive events, we have:

Also given that $A \cup B = S$, so:

By the addition rule of probability:

Let $P(A) = p$. Then $P(B) = 1 - p$, where $0 \le p \le 1$.

We wish to maximize the product:

Taking the derivative with respect to $p$ and setting it to zero:

Since $f^{\prime\prime}(p) = -2 < 0$, the function achieves its maximum at $p = 0.5$:

Thus, the maximum value of $P(A)P(B)$ is 0.25.