x, y and z are all unique numbers. If x is chosen randomly

Gmat_mission 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 and z is not?

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

[spoiler]OA=E[/spoiler]

Source: Veritas Prep

Are the numbers chosen with or without replacement?
For example, can y and z both be 23?

Gmat_mission wrote:x, y and z are all unique numbers.

To answer Brent's question, when they say x, y and z are "unique numbers", I'm guessing they're trying to say that the selection is being done without replacement. But it's mathematically wrong to present a probability question this way (and I missed their 'unique' comment the first time I read it too, since any real GMAT question would mention whether the selection was done with or without replacement), because when they say "x, y and z are all unique numbers", that suggests that x, y and z stand for specific (as yet unknown) values. You can't then ask a probability question as if we're choosing x, y and z randomly from a set, since that means x, y and z do not stand for specific values. They seem to want x, y and z to be what are called "random variables" in math, but those are well outside the scope of the GMAT.

Thanks Ian.
Not sure how I missed "unique." Thanks for clarifying.
Cheers,
Brent

Gmat_mission 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 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 is prime} 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