Question

Let S be sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(.) denotes the probability of the event, the maximum value of P(A) P(B) is _____.

Answer 1

Concept:

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint, that is, P(A ∩ B) = 0

Also P(S) = 1

Calculation

P (A ∪ B) = P(A)+P(B)+P(A∩B)

1 = P(A)+P(B)+ 0

∴ P(A) + P (B) = 1

Let P(A) = x ∴ P(B) = 1 – x

F(x) = P(A).P(B) = x (1 – x)

F(x) = x – x²

F’(x) = 1 – 2x

F’’(x) = -2 < 0

∴ F’(x) = 1 – 2x = 0   (gives max value of x)

1 – 2x = 0  

∴ x = ½

P(A) = ½ and P(B) = 1 – x = 1 – ½ = ½

P(A).P(B)  = ½ × ½  = ¼ = 0.25

The maximum value of P(A)P(B) is 0.25.

Important Point:

When sum is a constant, product of two numbers becomes maximum when they are equal.

∴ P(A) + P (B) = 1

P(A) = P(B) = ½