Title: Two dice are rolled. Find the probability that the s Post by: woot woot on Sep 17, 2014 Two dice are rolled. Find the probability that the sum was a 12 given that one of the numbers was a 6.
A.2/11 B.1/11 C.1/6 D.1/12 Title: Re: Two dice are rolled. Find the probability that the s Post by: bio_man on Sep 17, 2014 B.1/11 P(sum = 12 | one number is a 6) = P(sum = 12 and one number is a 6) / P(one of the numbers is a 6) = P(both numbers are 6) / (1 − P(neither number is a 6)) = (1/36) / (1 − (5/6)²) = (1/36) / (1 − 25/36) = (1/36) / (11/36) = 1/11 Alternate method: List all possible combinations when rolling 2 dice: 1−1 ..... 2−1 ..... 3−1 ..... 4−1 ..... 5−1 ..... 6−1 1−2 ..... 2−2 ..... 3−2 ..... 4−2 ..... 5−2 ..... 6−2 1−3 ..... 2−3 ..... 3−3 ..... 4−3 ..... 5−3 ..... 6−3 1−4 ..... 2−4 ..... 3−4 ..... 4−4 ..... 5−4 ..... 6−4 1−5 ..... 2−5 ..... 3−5 ..... 4−5 ..... 5−5 ..... 6−5 1−6 ..... 2−6 ..... 3−6 ..... 4−6 ..... 5−6 ..... 6−6 Of these, 11 combinations have at least one 6 1−6, 2−6, 3−6, 4−6, 5−6, 6−1, 6−2, 6−3, 6−4, 6−5, 6−6 Of these 11, only 1 has sum = 12 P(sum = 11 | one of the numbers = 6) = 1/11 —————————————————————————————— NOTE: If you had asked to find the probability that the sum was 12 given that the FIRST number was a 6, then answer would indeed be 1/6, since there are only 6 combinations where FIRST number = 6: 6−1, 6−2, 6−3, 6−4, 6−5, 6−6 and of these 6 combinations, only 1 has sum = 12 If you had asked to find the probability that the sum was 12 given that the SECOND number was a 6, then answer again would be 1/6, since there are only 6 combinations where SECOND number = 6: 1−6, 2−6, 3−6, 4−6, 5−6, 6−6 and of these 6 combinations, only 1 has sum = 12 So in both these cases, we are starting out with 6 combinations, and the probability that sum is 12 from these 6 combinations is 1/6 BUT, since you do not specify which number is 6, only that one of the numbers is a 6, we must look at all combinations where there is a 6. There are 11 such combinations (as I've shown above), and the probability that sum is 12 from these 11 combinations is 1/11 |