Question

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

Answer 1

It is given that $A$ and $B$ are mutually exclusive.
Also, it is given that $A\cup B=S$
This means $A$ and $B$ are collectively exhaustive.
Now if $A$ and $B$ are both mutually exhaustive and collectively exhaustive, then
$P\left(A\right)+P\left(B\right)=1\Rightarrow P\left(B\right)=1-P\left(A\right)$

Let $P\left(A\right)=x$ and $y=P\left(A\right)P\left(B\right)$
Then $y=x-x^2$
$\frac{dy}{dx}=1-2x=0\Rightarrow x=\frac{1}{2}\frac{d^{2}y}{d{x}^2}{|}_{x=1/2}=-2<0$
Therefore, maximum value of $y$ will occur at $x=\frac{1}{2}$
and $y_{max}=\frac{1}{2}-{\left(\frac{1}{2}\right)}^2=0.25$