History and rules
From knucklebones to fixed odds
Wagering is older than the mathematics that describes it by several thousand years, which is why the mathematics arrived as a shock.
AntiquityLots, knucklebones and marked dice appear across the ancient world, used for divination and for division of property as well as for play. The equipment is recognisable; the idea that its behaviour could be calculated is entirely absent.
Medieval periodDice games become common enough to attract repeated prohibition, usually on grounds of public order or of soldiers gambling away equipment. Card games arrive later and spread quickly once printing makes decks affordable.
Seventeenth centuryA practical question about how to divide the stake in an interrupted game is put to mathematicians. The correspondence that answers it establishes expectation as a calculable quantity.
Eighteenth centuryPublic draws are used as instruments of state finance. Coffee houses become venues where wagers on shipping, lives and public events are written down, priced and traded.
Nineteenth centuryOrganised racing produces the first systematic fixed-odds bookmaking, with prices posted, liabilities balanced across a book, and a margin built into the quoted odds.
Twentieth century onwardMachines, licensed venues and then networked systems industrialise the same propositions, and the regulatory apparatus described on the regulation page grows up around them.
Games of chance are among the oldest recorded human activities, and for most of their history nobody could calculate anything about them. Astragali, the ankle bones of sheep, were thrown for outcomes long before dice were manufactured. Lots were cast to allocate land, to select officials and to settle disputes, and the practice sat comfortably alongside a belief that the outcome was not random at all but decided elsewhere. That belief is worth taking seriously, because it explains why nobody counted. There is no reason to enumerate a sample space if you think the result is a verdict.
The problem of points
The decisive change came from an ordinary practical dispute, usually called the problem of points. Two players are partway through a game for a stake when the game must be abandoned. One is ahead. How should the stake be divided? The obvious answers, splitting evenly or awarding it to the leader, are both clearly wrong, and neither survives a moment of argument.
The insight that resolved it was to divide the stake not according to what had happened but according to what would have been likely to happen. That required enumerating the possible remaining sequences of play, counting how many led to each player winning, and dividing in proportion. It is exactly the procedure used everywhere else on this desk, and it was novel. Once expectation could be computed for an unfinished game, it could be computed for any wager, and the analysis of games of chance became a branch of mathematics rather than a matter of experience.
Coffee houses, ships and the birth of priced risk
The commercial consequence appeared quickly and in an unexpected place. Eighteenth-century coffee houses functioned as informal exchanges where merchants, shipowners and speculators met, and wagers written there covered shipping arrivals, harvests, lives and public events. Some of those wagers were recreational. Others were what would now be called insurance, distinguished from a bet only by whether the person taking it stood to lose something real if the event occurred.
That distinction is a legal one rather than a mathematical one, and drawing it took a long time. The same arithmetic prices both, and the institutional separation of insurance from wagering, which took most of a century and a good deal of legislation, is one of the most consequential outcomes of the whole history. Once insurable interest was required, the two activities diverged permanently.
Fixed odds and the book
Organised racing produced the next structural change. Where earlier wagering was largely a matter of two parties agreeing terms, a bookmaker offers prices on every runner simultaneously and accepts stakes from many parties at once. The prices are set so that the implied probabilities sum to more than one, and the excess is the margin. It is the same construction as the shortfall on a wheel payout, expressed as a set of prices instead of a single percentage.
Working, the shape of an over-round book
a price of 2 to 1 implies a probability of 1/3if every implied probability is summed across all runnersa fair book sums to exactly 1.00a commercial book sums above 1.00; the excess is the marginWhat that construction gave the bookmaker was not certainty on any single event but a portfolio. By adjusting prices as money arrives, liabilities can be balanced so the outcome matters less than the volume, which is the same principle that lets an insurer write policies without knowing which will claim. It is the commercial application of the law of large numbers, and it arrived before anyone described it in those terms.
What the history shows
Two threads run through the whole account. The first is that the games have barely changed: dice, wheels, cards and draws are ancient, and the modern versions differ mainly in manufacturing tolerance and supervision. The second is that everything else has changed completely. The propositions were unpriced, then priced; unregulated, then regulated; a matter of fortune, then a matter of arithmetic. The equipment on the table is old. The understanding of what it does is recent.