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

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

by Gmat_mission » Sun Jun 30, 2019 9:05 am

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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]

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by Brent@GMATPrepNow » Tue Jul 02, 2019 8:47 am
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]

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Are the numbers chosen with or without replacement?
For example, can y and z both be 23?
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by Ian Stewart » Tue Jul 02, 2019 10:15 am
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.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by Brent@GMATPrepNow » Tue Jul 02, 2019 2:55 pm
Thanks Ian.
Not sure how I missed "unique." Thanks for clarifying.
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by Scott@TargetTestPrep » Sat Jul 06, 2019 4:52 pm
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

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