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Dice and craps

Field: Game mechanics. This page was last modified on 17 August 2026.

Two dice generate thirty-six equally likely outcomes and eleven possible totals, and the uneven way those thirty-six outcomes distribute across the eleven totals is the foundation of every dice game. Craps is the most fully developed example: a game whose entire structure, including its best and worst wagers, can be derived from one small table.

The sample space

Two distinguishable dice produce 6 x 6 = 36 ordered outcomes, each with probability 1/36. Totals are not equally likely because different numbers of ordered outcomes produce them. A total of seven can be made six ways, six and eight five ways each, five and nine four ways, four and ten three ways, three and eleven two ways, and two and twelve one way each.

Treating the dice as distinguishable is essential even though they look identical in play. A three and a four is two outcomes, not one, and a pair of threes is only one. Forgetting this is the classic beginner's error and it produces a sample space of twenty-one rather than thirty-six.

Point probabilities

Once a point number is established in craps, only two totals matter: the point and seven. Every other roll is irrelevant, so the probability of making the point before a seven is simply the point's share of the ways that resolve the bet.

Deriving the pass line

The pass line wins immediately on seven or eleven, which is 8/36 of first rolls, and loses immediately on two, three or twelve, which is 4/36. The remaining 24/36 establish a point, and each point then resolves at the probability in the table above.

Summing the point branches gives a total win probability of 244/495, or 0.4929. Because the bet pays even money, the expected value per unit is 2 x 0.4929 - 1 = -0.0141, an edge of 1.41%. The don't pass bet is the near-mirror image, with the twelve turned into a push rather than a win, which is exactly the adjustment that prevents it from being a favourable bet; its edge is 1.36%.

Odds bets and the shape of the table

Craps also allows a supplementary wager, taken after a point is established, that pays at the true odds in the table above. Because it is paid correctly, it carries no edge at all. It cannot be made on its own, so it does not create a favourable game; what it does is dilute the edge on the money already committed, which is why the combined figure for a line bet backed by this supplement is much lower than 1.41%.

At the other end of the table sit the single-roll propositions in the centre of the layout. A bet on seven alone wins 6/36 of the time, so the true odds are 5 to 1, and it is commonly paid 4 to 1: an edge of 16.67%. The best and the worst wagers in the building are frequently within arm's reach of each other on the same layout.

Dice, fairness and control

Casino dice are made to tight tolerances with flush, filled pips precisely so that the sample space above is the real one; a die with drilled, unfilled pips is very slightly unbalanced by the missing material. Claims of controlled throwing, in which a shooter influences the outcome enough to overcome the edge, have never been demonstrated under conditions that separate the claim from ordinary scatter, and the required effect size is large: the pass line edge would have to be erased by a measurable shift in the frequency of sevens.