Two Dice, Eleven Totals: The Probability Behind Every Roll

Roll two standard six-sided dice and add the results, and you get a total between 2 and 12. Those eleven totals are not equally likely, which surprises many people on first meeting the idea. The reason is simple counting, and understanding it makes dice games much easier to read.

Counting the combinations

Each die has six faces, so two dice can land in 6 times 6, or 36, equally likely combinations. Treat the dice as distinguishable, say one red and one blue, so that a red 1 with a blue 2 is a different outcome from a red 2 with a blue 1. Each of the 36 combinations has the same chance, one in 36.

To find the probability of a total, count how many combinations produce it and divide by 36.

The full table

Here is how many combinations make each total:

Total 2: 1 way (1 and 1).
Total 3: 2 ways.
Total 4: 3 ways.
Total 5: 4 ways.
Total 6: 5 ways.
Total 7: 6 ways.
Total 8: 5 ways.
Total 9: 4 ways.
Total 10: 3 ways.
Total 11: 2 ways.
Total 12: 1 way.

Add them up: 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 5 plus 4 plus 3 plus 2 plus 1 equals 36. The counts check out.

Why seven is special

Seven has the most combinations, six of them: 1 and 6, 2 and 5, 3 and 4, and the reverse of each. Its probability is 6 in 36, or one in six, about 16.7 percent. The extremes, 2 and 12, each appear only once in 36 rolls on average, about 2.8 percent.

This pattern is why the number seven plays such a central role in games like craps, which we cover in our guide to craps table terms. Totals near the middle are much more common than those at the edges.

Putting the table to work

With the counts in hand, you can answer many practical questions:

Chance of 7 or 11 on a roll: 6 plus 2 equals 8 combinations out of 36, or 22.2 percent.
Chance of 2, 3, or 12: 1 plus 2 plus 1 equals 4 out of 36, or 11.1 percent.
Chance of rolling doubles: 6 out of 36, or one in six.
Chance of a total of 6 or 8: 5 plus 5 equals 10 out of 36, about 27.8 percent.

Check your own sums by making sure that the complementary outcomes add up to 36.

From probability to payouts

A fair payout would match the odds. If a total has a probability of 1 in 6, fair odds are 5 to 1, since for every one win there are five losses on average. If a game pays less than that, the difference is the house's edge. This is the same logic as in our article on expected value.

For example, a bet on a total of 12 has a 1 in 36 chance, so fair odds would be 35 to 1. A game that pays 30 to 1 would be offering less than fair odds, giving the house an advantage of about 5 divided by 36, or roughly 14 percent. The payout looks large, but the cost per dollar is high.

Common misunderstandings

Two mistakes appear frequently:

Believing that all totals are equally likely because there are eleven of them. They are not, because the combinations differ in number.
Believing that a total is due after it has not appeared for some time. The dice have no memory, and each roll is independent.

A quick exercise

Try working out the probability of getting a total of at least 10. Totals 10, 11, and 12 have 3, 2, and 1 ways respectively, so that is 6 combinations out of 36, or one in six, about 16.7 percent. It matches the chance of rolling a seven, which is a neat coincidence worth remembering.

Dice are a wonderful way to learn probability. If you play any game involving them for money, set a firm budget first. Gambling is for adults only, with legal ages that vary by location.

Frequently asked questions

Why are there 36 outcomes and not 21? Counting only distinct pairs gives 21, but those are not equally likely. Treating the dice as distinguishable gives 36 equally likely outcomes, which is the correct basis for probabilities.

What is the least likely total? Totals 2 and 12 are tied, each with a single combination.

Does a weighted or worn die change this? Yes, but regulated casino dice are manufactured and inspected to keep outcomes fair.