GATE | GATE-CS-2014-(Set-3) | Question 58

Let S be the sample space and two mutually exclusive events A and B be such that A U B = S. If P(.) denotes the probability of the event. The maximum value of P(A)P(B) is ______
(A) 0.5
(B) 0.25
(C) 0.225
(D) 0.125


Answer: (B)

Explanation: 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, lets 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.

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

Hence option B is the correct choice.

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