How this calculator works
Each fair six-sided die shows one of six faces with probability 1/6, and the two dice act independently. Treating the dice as distinct gives 6 x 6 = 36 equally likely ordered pairs, such as (1,4) and (4,1). Counting how many pairs add to a target gives the number of ways, and dividing by 36 gives the probability. Sums near the middle have more ways because several different pairs reach them, while 2 and 12 have only one way each. The table also shows cumulative probabilities: at most a sum includes every total up to that sum, and at least a sum includes every total from that sum upward.
Formula
P(sum = s) = ways(s) / 36, where ways(s) = s - 1 for s from 2 to 7 and ways(s) = 13 - s for s from 8 to 12. The expected sum is 7. For a range, add the ways of every sum in the range and divide by 36.
Worked example
A total of 7 has six ways: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Its probability is 6/36 = 1/6. A total of 12 has only the pair (6,6), so its probability is 1/36. At least 10 means 10, 11 or 12, which have 3, 2 and 1 ways, so the probability is (3 + 2 + 1) / 36 = 6/36 = 1/6.
Advertisement
Frequently asked questions
Why is seven the most likely sum?
Seven has six ordered pairs that reach it, more than any other total. The counts rise from one way at 2 to six ways at 7 and then fall symmetrically to one way at 12.
Do the dice have to be different colours?
No. Treating the dice as distinct is just a counting method. It ensures that (1,6) and (6,1) are counted as two outcomes, which is correct because each pair is equally likely.
Are at least and at most inclusive?
Yes. At least 10 includes 10, 11 and 12. At most 4 includes 2, 3 and 4. Exactly includes only the target sum.
Can this handle loaded or uneven dice?
No. The page assumes two identical fair six-sided dice. Loaded dice need separate face probabilities, and dice with other numbers of sides need a general dice probability model.