Dice Roll Difference Probability

Question

Two people each rolling their own pair of 6 sided dice, then comparing the difference in the totals they rolled. How would one calculate the probability of all the differences they could have? (As in, the probability their rolls have a difference of 0, 1, 2... and 10)

Answer 1

Hi John,

We can calculate the probability of each sum from 2 to 12 for each pair of dice rolls. Each die roll is an independent event with 6 different possible outcomes, so each pair of dice rolls can have one of 36 different outcomes (6 x 6). Of these 36 outcomes, only one outcome produces a sum of 2 (rolling two 1's) and only one outcome produces a sum of 12 (rolling two 6's). Sums of 3 (2 and 1, or 1 and 2) and 11 (6 and 5, or 5 and 6) are produced by 2/36 or 1/18 possibilities each. Sums of 4 and 10 have a probability of 3/36 or 1/12 each; 5 and 9 have a probability of 4/36 or 1/9 each; 6 and 8 have a probability of 5/36 each; and a sum of 7 has a probability of 6/36 or 1/6.

We can then calculate the probability of the difference between a pair of dice rolls as follows:

For the difference between the sums of each pair of rolls to be 10, one must roll a 12 and the other must roll a 2. There is a 1/36 chance of each of these happening. We can then calculate that there is a (1/36) x (1/36) chance for the first person to roll a 12 and the second person to roll a 2. This gives us a probability of 1/1296, which we double to account for the possibility of the first person rolling a 2 and the second person rolling a 12 to reach a final probability of 1/648 for the sum of two pairs of dice rolls to have a difference of 10.

The difference of 10 is the simplest case, but the same logic can be applied to find probabilities for the rest of the possible differences.

I hope this helps.