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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Three friends Alan, Roger and Peter ##### This topic has 2 expert replies and 1 member reply ### Top Member ## Three friends Alan, Roger and Peter Three friends Alan, Roger and Peter attempt to answer a question on an exam. Alan randomly guesses the answer, giving him a 1/5 probability of guessing correctly. Roger cheats by looking at the paper of the student in front of him, giving him a 2/3 probability of answering correctly. And Peter dutifully performs the calculations, then marks the answer, giving him a 5/6 probability of a correct answer. What is the probability that the question is answered correctly, but not via cheating? A 1/18 B 1/9 C 23/90 D 5/18 E 13/45 Can some experts help me find the solution in this? OA E ### GMAT/MBA Expert Legendary Member Joined 14 Jan 2015 Posted: 2667 messages Followed by: 122 members Upvotes: 1153 GMAT Score: 770 Top Reply lheiannie07 wrote: Three friends Alan, Roger and Peter attempt to answer a question on an exam. Alan randomly guesses the answer, giving him a 1/5 probability of guessing correctly. Roger cheats by looking at the paper of the student in front of him, giving him a 2/3 probability of answering correctly. And Peter dutifully performs the calculations, then marks the answer, giving him a 5/6 probability of a correct answer. What is the probability that the question is answered correctly, but not via cheating? A 1/18 B 1/9 C 23/90 D 5/18 E 13/45 Can some experts help me find the solution in this? OA E This question could use some cleaning up. It seems to be asking what the probability is that at least one of the friends answers the question correctly, so long as that friend is not Roger, the cheater. (It wouldn't make much sense to assume that the friends are working together and answering the problem as a cohesive unit.) If so, this could happen one of three ways: 1) Alan and Peter both answer correctly, and Roger is wrong. 2) Alan answers correctly, and Peter and Roger are wrong 3) Peter answers correctly, and Alan and Roger are wrong Scenario 1: P(Alan correct) = 1/5; P(Peter correct)= 5/6; P(Roger wrong) = 1/3 --> 1/5 *5/6 * 1/3 = 1/18 Scenario 2: P(Alan correct) = 1/5; P(Peter wrong)= 1/6; P(Roger wrong) = 1/3 --> 1/5 *1/6 * 1/3 = 1/90 Scenario 3: P(Peter correct) = 5/6; P(Alan wrong)= 4/5; P(Roger wrong) = 1/3 --> 5/6 * 4/5 * 1/3 = 2/9 Add the scenarios together: 1/18 + 1/90 + 2/9 = 5/90 + 1/90 + 20/90 = 26/90 = 13/45. The answer is E _________________ Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course

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### GMAT/MBA Expert

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lheiannie07 wrote:
Three friends Alan, Roger and Peter attempt to answer a question on an exam. Alan randomly guesses the answer, giving him a 1/5 probability of guessing correctly. Roger cheats by looking at the paper of the student in front of him, giving him a 2/3 probability of answering correctly. And Peter dutifully performs the calculations, then marks the answer, giving him a 5/6 probability of a correct answer. What is the probability that the question is answered correctly, but not via cheating?

A 1/18
B 1/9
C 23/90
D 5/18
E 13/45
Let A, R, P = the probability that Alan, Roger and Peter correctly answer the question, respectively. Then the probability the question is answered correctly but not by cheating is the probability that Roger (the cheater) answers it incorrectly but either Alan or Peter or both answer it correctly.

P(A, not R, P) = 1/5 x 1/3 x 5/6 = 5/90
P(A, not R, not P) = 1/5 x 1/3 x 1/6 = 1/90
P(not A, not R, P) = 4/5 x 1/3 x 5/6= 20/90

Thus the probability is 5/90 + 1/90 + 20/90 = 26/90 = 13/45.

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### Top Member

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DavidG@VeritasPrep wrote:
lheiannie07 wrote:
Three friends Alan, Roger and Peter attempt to answer a question on an exam. Alan randomly guesses the answer, giving him a 1/5 probability of guessing correctly. Roger cheats by looking at the paper of the student in front of him, giving him a 2/3 probability of answering correctly. And Peter dutifully performs the calculations, then marks the answer, giving him a 5/6 probability of a correct answer. What is the probability that the question is answered correctly, but not via cheating?

A 1/18
B 1/9
C 23/90
D 5/18
E 13/45

Can some experts help me find the solution in this?

OA E
This question could use some cleaning up. It seems to be asking what the probability is that at least one of the friends answers the question correctly, so long as that friend is not Roger, the cheater. (It wouldn't make much sense to assume that the friends are working together and answering the problem as a cohesive unit.)

If so, this could happen one of three ways:

1) Alan and Peter both answer correctly, and Roger is wrong.
2) Alan answers correctly, and Peter and Roger are wrong
3) Peter answers correctly, and Alan and Roger are wrong

Scenario 1: P(Alan correct) = 1/5; P(Peter correct)= 5/6; P(Roger wrong) = 1/3 --> 1/5 *5/6 * 1/3 = 1/18
Scenario 2: P(Alan correct) = 1/5; P(Peter wrong)= 1/6; P(Roger wrong) = 1/3 --> 1/5 *1/6 * 1/3 = 1/90
Scenario 3: P(Peter correct) = 5/6; P(Alan wrong)= 4/5; P(Roger wrong) = 1/3 --> 5/6 * 4/5 * 1/3 = 2/9

Add the scenarios together: 1/18 + 1/90 + 2/9 = 5/90 + 1/90 + 20/90 = 26/90 = 13/45. The answer is E
Thanks a lot!

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