Question

When a pair of dice is rolled, what is the probability that the sum of the dice is 5, given that the outcome is not 6? The probability that the sum of the dice is 5 given that the outcome is not 6 is (Type an integer or a simplified fraction.)

Answer 1

By definition of conditional probability,

P(sum is 5 | sum is not 6) = P(sum is 5 AND sum is not 6) / P(sum is not 6)

There are 4 ways of rolling the dice to get a sum of 5 out of 36 possible outcomes:

(1, 4), (2, 3), (3, 2), (4, 1)

so P(sum is 5) = 4/36 = 1/9.

There are 5 ways of getting a sum of 6 out of 36 possible outcomes:

(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)

hence 31 ways of *not* getting a sum of 6, so P(sum is not 6) = 31/36.

The sum is 5 and *not* 6 simultaneously for the 4 ways already listed above, so the intersection of these two events is those 4 ways, which means

P(sum is 5 AND sum is not 6) = 1/9

Then

P(sum is 5 | sum is not 6) = (1/9) / (31/36) = 4/31