Which of the following statements is correct given that the events a and b have nonzero probabilities? a. a and b can be both independent and mutually exclusive b. a and b cannot be both independent and mutually exclusive c. a and b are always independent d. a and b are always mutually exclusive

The statement which is correct is:

B.     A and B cannot be both independent and mutually exclusive.

Let we toss a coin single time.

and event A denote the tossing up a head.

and event B denote tossing a tail.

                               P(A∩B)=0

                         P(A∩B)=P(A)×P(B)

Here we have:

P(A)≠0 and P(B)≠0

So, if the events are independent then we have:

P(A∩B)≠0

Hence, the events can't be mutually exclusive.

Similarly if the two events are mutually exclusive then they can't be independent.

i.e. both mutually exclusive property and independent property can't exist at the same time if the two events have non-zero probability.

     Hence, the answer is: Option: B

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