6.54.

For any three events A, B, and C, show that

Hint: Write A ? B ? C = A ? ( B ? C) and then apply Eq. (6.8).

6.55.

Given that P( A) = 0.9, P( B) = 0.8, P( A ? B) = 0.75, find ( a) P( A ? B); ( b) P( A ? B); and ( c) P( A ? B)

6.56.

Show that if events A and B are independent, then

Hint: Use Eq. (6.102) and the relation

6.57.

Let A and B be events defined in a sample space S. Show that if both P( A) and P( B) are nonzero, then the events A and B cannot be both mutually exclusive and independent.

Hint: Show that condition (6.21) will not hold.

6.58.

A certain computer becomes inoperable if two components A and B both fail. The probability that A fails is 0.01, and the probability that B fails is 0.005. However, the probability that B fails increases by a factor of 3 if A has failed.

1. Calculate the probability that the computer becomes inoperable.

2. Find the probability that A will fail if B has failed.

6.59.

A certain binary PCM system transmits the two binary states