Two Dice Probability Calculator

Two-dice probability is the study of the totals that two independent fair six-sided dice produce. Because every ordered pair is equally likely, the subject reduces to counting, and the count behind the totals is the familiar 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.

That simplicity is the point. A question that looks like it needs a formula usually needs one division: count the arrangements, divide by 36. Loaded dice are the interesting variation, because they remove the equal-likelihood assumption and force a convolution instead.

Each die shows one of six faces, so the pair shows 36 ordered outcomes, all with probability 1/36. A total of k is reached by however many pairs add to k: one pair for 2, two for 3, and so on up to six for 7.

A threshold is handled by adding counts on one side of it. The lower tail is the running sum, and the upper tail is one minus the running sum up to k minus 1, which keeps the threshold total in exactly one of the two tails.

A named pair is a count of arrangements rather than of totals. Two different faces give two arrangements when order is free and one when it is fixed, while two matching faces give one either way.

With loaded dice the outcomes are no longer equally likely. Each face carries a normalised weight, and the probability of a total is the sum over the pairings that reach it. Summing every total recovers one, which is the check that the weights were read correctly.

The count behind a total is the whole result. Six arrangements for 7, one for 2, and the same one for 12 explain the entire ranking without any further calculation.

The odds-against row is the same fact inverted: 30 to 6 for a total of 7, 35 to 1 for a total of 2. Odds against are easier to read than probabilities once the numbers get small.

The expected waiting time is one over the probability, so a total of 7 arrives about every 6 rolls and a total of 2 about every 36. This is a long-run average, not a promise about the next roll.

In weighted mode the useful comparison is against the fair-dice value in the summary. A loaded die that leans low raises the low totals and lowers the high ones, and the change-from-fair row quantifies it for the total you asked about.

• Which total you ask about. The counts run from 1 at the edges to 6 in the middle, so the total matters more than anything else.
• Whether the dice are fair. Loaded dice leave the 11 possible totals intact but redistribute their probability.
• Whether order matters, in pair mode. It doubles the chance for two different faces and changes nothing for two matching faces.
• How many rolls you simulate. Sampling noise falls as 1 over the square root of the count, so 20,000 rolls leaves gaps of a few tenths of a percent.

• Settling arguments about dice in games, craps, or tabletop rules.
• Checking a loaded-dice design where the weights are known but the resulting distribution is not.
• Teaching why the totals of two dice are not equally likely, which is the standard first lesson in probability.
• Estimating waiting times, such as how many rolls a craps pass or a Yahtzee attempt needs.

• The two dice are assumed independent. Dice that share a mechanism, such as a weighted pair sold as a set, can break that.
• Fair dice are assumed uniform over the six faces. Loaded dice handled in weighted mode are only as accurate as the weights you enter.
• Waiting times are geometric averages over independent rolls, so the actual gap between two appearances of a total can be much longer than the average.
• The simulation uses a simple linear congruential generator, which is reproducible and unbiased enough for a demonstration but is not a cryptographic or research-grade source of randomness.