Sample Space(S) – A set of all possible outcomes/events of a random experiment.

Mutually Exclusive Events – Those events which can’t occur simultaneously.

P(A)+P(B)+P(A∩B)=1 Since the events are mutually exclusive, P(A∩B)=0.

Therefore, P(A)+P(B)=1

Now, we know that AM >= GM So, (P(A)+P(B))/2 >= sqrt(P(A)*P(B)) P(A)*P(B) <= 1/4

Hence max(P(A)*P(B)) = 1/4.

We can think of this problem as flipping a coin, it has two mutually exclusive events ( head and tail , as both can’t occur simultaneously).

And sample space S = { head, tail } Now, let’s say event A and B are getting a “head” and “tail” respectively.

Hence, S = A U B. Therefore, P(A) = 1/2 and P(B) = 1/2.

P(A).P(B) = 1 /4 = 0.25.