Question

A pair of events a and b cannot be simultaneously mutuallyexclusive and independent. prove that if p(a)>0 and p(b)>0,then:

if a and b are mutually exclusive, they cannot beindependent.

Answer 1

Mutually exclusive events cannot occur simultaneously, meaning P(A AND B) = 0, while independent events have no effect on each other's occurrence, and P(A AND B) should equal P(A)P(B). Since P(A) and P(B) are greater than zero, they cannot be both mutually exclusive and independent.

Step-by-step explanation:

If two events A and B are mutually exclusive, this means that they cannot occur at the same time. The definition of mutually exclusive events implies that P(A AND B) = 0.

However, if A and B were also independent, it would mean that the occurrence of A does not affect the probability of B occurring and vice versa, which leads to the formula P(A AND B) = P(A)P(B).

Given that P(A)>0 and P(B)>0, if A and B were independent, then P(A)P(B) would be a positive number, since the product of two positive numbers is positive.

This contradicts the definition of mutually exclusive, which requires that P(A AND B) = 0.

Therefore, it's impossible for events A and B to be both mutually exclusive and independent when both have a non-zero probability of occurring.