Probability of Rolling 3 With Two Dice: All Ordered Outcomes

The answer under a fair-dice model

For two independent fair six-sided dice, the probability of a total of 3 is 2/36 = 1/18, or approximately 5.5556%. Each die has six equally likely faces, giving 6 × 6 = 36 equally likely ordered pairs.

This calculation assumes faces numbered 1 through 6 and independent results. Loaded dice, a different number of faces or a rule that rerolls certain outcomes creates a different model.

List the successful outcomes

The complete list is (1, 2), (2, 1). There are 2 successful ordered pairs.

The first coordinate records die A and the second records die B. For a nonmatching pair, reversing the coordinates represents another outcome. Matching pairs such as (3, 3) occur only once. Enumerating ordered pairs prevents inconsistent counting.

Why the eleven possible totals are not equally likely

Totals run from 2 to 12, but the paths to each total differ. A total of 7 has 6 ordered outcomes, giving 6/36. Compare that count with the 2 outcomes for 3 rather than assigning every total a probability of 1/11.

The sample space consists of pairs of faces, not just the labels of their sums. Whenever several equally likely outcomes collapse into one label, count how many outcomes belong to each label.

Expected counts are not a schedule

Across 360 independent rolls, the expected count of sum 3 is 360 × 2/36 = 20. Actual counts can differ. “Expected” describes the average across many imagined repetitions, not a promise for the next block of rolls.

If the last ten rolls did not total 3, the probability on the next roll remains 2/36 under the independent model. Previous misses do not create a debt that the dice must repay.

A related question: two successes in a row

The probability that two consecutive rolls both total 3 is (2/36) × (2/36) = 1/324, approximately 0.3086%. Multiply because both events must happen and the rolls are independent.

The probability of not getting 3 on one roll is 1 − 1/18 = 17/18. Those two complementary probabilities add to 1. Always distinguish “both,” “at least one” and “exactly one” before selecting an operation.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.