sachin_yadav wrote:x, y and z are all unique numbers. If x is chosen randomly from the set {7,8,9,10,11} and y and z are chosen randomly from the set {20,21,22,23}, what is the probability that x and y are prime numbers and z is not ?

(A) 1/5
(B) 3/20
(C) 13/20
(D) 3/10
(E) 1/10

Since x is chosen from a set that is different from y and z, we see that the event x is prime is independent from the events y is prime and z is not prime. However, the events y isprime and z is not prime are not independent since they are chosen from the same set. Recall that if two events A and B are independent, then P(A and B) = P(A) x P(B). However, if A and B are not independent, then P(A and B) = P(A) x P(B|A) where P(B|A) means the probability of B given that A has occurred. Thus:

P(x is prime) = 2/5

and

P(y is prime and z is not prime) = P(y is prime) x P(z is not prime, given that y is prime) = 1/4 x 3/3 = 1/4

Finally, we have P(x is prime and (y is prime and z is not prime)) = 2/5 x 1/4 = 2/20 = 1/10.

Answer: E