Dice, wheels and the layout

Two six-sided dice produce thirty-six equally likely ordered results, and those thirty-six map onto eleven possible totals with very unequal frequencies. A total of seven occurs six ways, six and eight occur 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. The triangular shape of that distribution is the foundation of every dice game ever designed.

It is also the source of the most persistent misunderstanding in the subject. The totals are not equally likely, but the individual ordered results are, and conflating the two produces a long list of wrong conclusions. Anyone working through a dice paytable should start by writing out the thirty-six ordered results, because every probability in the game is a count of that grid.

A wheel game's mathematics is entirely determined by how many pockets it has and how many of them a given bet covers. On a thirty-seven-pocket wheel a single-number bet covers one pocket, and paying thirty-five to one against true odds of thirty-six to one produces the 2.70 per cent expectation derived earlier on this board. Add a second zero pocket, making thirty-eight, and the same thirty-five to one payout now sits against true odds of thirty-seven to one, and the expectation roughly doubles to about 5.26 per cent.

That single design change—one extra pocket, no change to any published payout—is the clearest illustration available of where a margin actually lives. It lives in the ratio between the number of ways to win and the multiple paid, and in nothing else. Two wheels that look nearly identical and pay identically can differ in expected cost by a factor of two.

A wheel layout offers a large variety of bets: single numbers, pairs, rows, columns, halves, colours. On a single-zero wheel almost all of them return the same 2.70 per cent expectation, because each is priced by the same rule—the payout equals the true odds computed as though the zero were absent. The variety is real as an experience and illusory as mathematics: the bets differ enormously in variance and not at all in expectation.

This is a general principle of game design worth carrying to other games. Where a game offers many options at a uniform margin, the options are shaping the distribution of outcomes rather than their average. Where a game offers options at differing margins, the differences are usually concentrated in the propositions that look most exciting, which on many dice and wheel layouts carry noticeably worse expectations than the plain bets beside them.