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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 $P(.)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is_____.

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Let

P(A) = x, P(B) = 1-x  ($\because$ A and B are mutually exclusive and A $\cup$ B = S)

We want to find max value of

f(x) = P(A)*P(B) = x*(1-x) = x – $x^{2}$

we find critical points i.e $f^{'}(x)$ = 0

$f^{'}(x)$ = 1-2x = 0

$\therefore$ x = ½ , this x can be minima or maxima, so we will find second derivative and if  $f^{''}(x)$ < 0, then x will give max value for $f(x)$

$f^{''}(x)$ = -2 < 0

$\therefore$ at x = ½ we get max value

 =x(1-x) = ½ * ½ = ¼ = 0.25
Answer:

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