Question

Prove: If A and B are mutually exclusive, and A and B are also independent, then either A or B has probability zero.

Answer 1

Given:
Events A and B are mutually exclusive.

So,

P(A AND B) = 0                                 (1)

because if Events A and B are mutually exclusive, it is impossible for them to happen together.

Events A and B are independent

So,

P(A AND B) = P(A) X P(B)             (2)

because if Events A and B are independent, the occurrence of one event has no effect on the occurrence of the other event.

Since LHS of equations (1) & (2) are same, equating RHS of (1) & (2), we get:

P(A) X P(B) = 0

This is product of 2 numbers with 0.

Thus, one of them should be 0.

This proves the required result: either A or B has a probablity 0.