Wheel and dice

Thirty-six outcomes, and everything that follows from them

Two dice generate a small, completely enumerable sample space. Every dice wager ever written is a rearrangement of the same eleven rows.

Fanfeud desk note. Figures on this page are worked out from the rules stated beside them.

A single fair die has six equally likely faces. Two dice thrown together have thirty-six equally likely ordered outcomes, because each of the six faces on the first die can be paired with each of the six on the second. The totals those outcomes produce are not equally likely, and the table above is the whole of the game's underlying structure. A seven can be made six ways, a two only one, and every dice proposition in existence is a claim about those counts.

It is worth noticing how much follows immediately. The chance of a seven on any one throw is six in thirty-six, or one in six. The chance of a total of four is three in thirty-six, or one in twelve. Because the dice do not remember, each throw restarts the same table from the top.

The pass line

The main dice wager resolves in two stages. On the first throw, a seven or an eleven wins immediately, while a two, three or twelve loses immediately. Any other total becomes the point, and from then on the wager wins if that point is repeated before a seven appears, and loses if the seven comes first.

The second stage is a race between two totals, and its probability follows from the counts alone. If the point is six, there are five ways to make a six and six ways to make a seven, so the point wins five times in eleven. The same logic gives four in ten for a five and three in nine for a four.

The free odds bet, and why it is genuinely different

Once a point is established, the rules of the game permit an additional wager on that point that is paid at its true odds: two to one on a four or ten, three to two on a five or nine, and six to five on a six or eight. Because the payout equals the true odds exactly, the expected value of that wager is zero.

This is the clearest demonstration on the whole desk that an edge is a consequence of payout schedules and nothing else. Remove the shortfall and the edge vanishes. It also explains why free odds are always attached to a wager that does carry an edge, and are capped as a multiple of it. The pairing keeps the combined proposition profitable to the game while allowing the second part to be paid honestly.

The rest of the layout

The remaining dice propositions are simply other totals with other payout schedules, and each is worked out the same way. A place bet on the six pays seven to six against true odds of six to five. A one-roll bet on any seven pays four to one against true odds of five to one, which is why it is among the most expensive lines in the whole family of games.

The spread between 1.41 per cent and 16.67 per cent on the same felt, decided purely by which box a chip is placed in, is the reason a dice layout is a good illustration of game design. Every one of those numbers was chosen. None of them emerged.