sacstat mark: a grid of pockets with a single one filledsacstat outcomes = 36

sacstat › Wheel and dice games › dice and craps

dice and craps

outcomes = 36

Two dice produce thirty-six equally likely ordered outcomes and eleven unequally likely sums. Nearly everything interesting about dice games comes from that mismatch, including the one bet in common use whose expected value is exactly zero.

The thirty-six cell grid

Enumerating two dice as a six-by-six grid removes all ambiguity. There is one way to make two and one to make twelve, six ways to make seven, and a tidy symmetric distribution in between. The grid is the source of every dice figure; nothing here requires memorising a table of probabilities.

A six by six grid of the thirty-six outcomes of two dice with each cell labelled by its sum, and the six cells that total seven highlighted along the diagonal.
Seven occupies six of thirty-six cells; two and twelve occupy one each.

The line bet

A pass line wager wins immediately on seven or eleven, loses immediately on two, three or twelve, and otherwise establishes a point that must be repeated before a seven appears. The point phase is a race between two fixed probabilities, which makes the whole computation a sum of a few simple terms.

The result is a win probability of 244/495, a shade under one half. The house edge is one minus twice that, or 7/495, which is about 1.41% — among the lowest edges on a casino floor, and a good illustration that a low edge is not the same as a good deal once round rate is taken into account.

Pass line, worked
come-out:  win  7,11        8/36
           lose 2,3,12      4/36
           point 4,5,6,8,9,10 remainder

point phase: p(repeat point before a 7) = ways(point) / (ways(point) + 6)

  point 4 or 10:  3/36 x 3/9  = 0.0278 each
  point 5 or 9:   4/36 x 4/10 = 0.0444 each
  point 6 or 8:   5/36 x 5/11 = 0.0631 each

p(win) = 8/36 + 2(0.0278 + 0.0444 + 0.0631) = 244/495 = 0.49293
edge   = 1 - 2 x 0.49293 = 7/495 = 1.414%

Free odds: the zero-edge wager

Once a point is established, the rules permit an additional wager paid at the true odds of repeating that point — 2 to 1 on four or ten, 3 to 2 on five or nine, 6 to 5 on six or eight. Because the payout equals the true price, its expected value is exactly zero. It is not a concession or a promotion; it is a structural feature that lets a player enlarge a position without enlarging the margin paid on it.

The consequence is arithmetic rather than strategic: the edge on the combined position falls as the odds portion grows, because the fixed cost of the line bet is spread across a larger total. Nothing is gained; the same expected loss is simply divided by a bigger number, and the variance of the position rises sharply in exchange.

True odds paid on the free odds wager
pointways to make itways to make 7true oddspayout
4 or 10362 to 12 to 1
5 or 9463 to 23 to 2
6 or 8566 to 56 to 5

Proposition bets

The centre of a craps layout carries single-roll wagers on specific sums, and their edges are far higher than the line — often ten to seventeen percent — for the same reason as everywhere else: the payout is written short of the true price. Any of them can be checked in one line against the thirty-six cell grid, which is the practical value of having drawn the grid at all.

Commonly misread. A low edge on the line does not spread to the rest of the table. Craps is not one game with one price; it is a collection of separate wagers sharing a pair of dice, and their expected values differ by more than an order of magnitude.

Return to the Wheel and dice games index on the reference landing.

‹ roulette card games ›

Last modified 17 August 2026.