If two fair dice are​ rolled, find the probability that the sum of the dice is 8​, given that the sum is greater than 3.

Answer 1

Sum is greater than 3.

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

There are 33 combinations whose sum is greater than 3 as shown above in bold italic.

Of those, 5 of them have a total of 8, so 5/33

Now using conditional probability:

Prob(>3) = 33/36 = 11/12

Prob(8) = 5/36

Prob ( 8 | >3) = Prob ( 8 AND >3) / Prob(>3) <--- def. of conditional probability

= 5/36 divided by 11/12

= 5/36 * 12/11

= 5/3 * 1/11

= 5/33

Answer 2

If we have 2 fair dice, we know the possible outcomes when added together are:

2 3 4 5 6 7 8 9 10 11 12

Each die by itself can roll 1-6, so if we multiply the total outcomes of each die, that gives us 36 possible combinations.

Since we're looking for an 8, we focus on the outcomes that give us an 8, namely

2 and 6

3 and 5

4 and 4

5 and 3

6 and 2

5 possible outcomes out of 36, which translates to 5/36, or 13.89%.

HOWEVER! No matter what we roll, our answer must be greater than 3. So outcomes that give us 2 or 3 must be removed from our final answer. The combinations of (1 and 1), (1 and 2) and (2 and 1) cannot be included, so we are left with 33 valid outcomes. Therefore:

5 outcomes out of 33 = 5/33, or 15.15%