Lotteries

Draw games: counting selections, and counting centuries

A draw game is the purest combinatorial object in the whole family, and also the one with the longest continuous institutional history.

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

A draw game asks a player to select a set of numbers, then draws a set and compares. Because the order in which numbers are drawn is irrelevant, the count of possible outcomes is a count of combinations rather than of arrangements, and the formula for combinations is the only tool needed.

The division by seven hundred and twenty is the entire reason the number is as small as it is, and it is also the step most often skipped. Ten billion ordered draws collapse to fourteen million unordered ones, because the game does not care what sequence the balls emerge in.

Why a small format change moves the number so far

Draw formats look similar and are not. Adding ten numbers to the pool while keeping the selection at six more than triples the number of combinations. This is why formats are periodically altered: the size of the pool is the primary control on how often a top prize is won, and therefore on how large the pool of unwon money can grow.

Expected value, and what a rollover really does

Unlike a wheel, a draw game has no fixed payout schedule for its top prize. The top prize is a share of a pool, and the pool changes between draws. That makes expected value a moving quantity rather than a constant, which is unusual in this family of games and frequently misunderstood.

The general shape is easy to state. Expected value per entry is the sum, over every prize tier, of the prize multiplied by the chance of winning it, less the cost of the entry. When a top prize is not won it is commonly added to the next draw, raising that prize without raising the odds against winning it. The expected value of an entry therefore rises after a rollover.

The counterweight matters and is often ignored. A larger advertised prize attracts more entries, which raises the chance that the winning selection is held by more than one person, and the top prize is then divided. Popular selections make this worse: sequences, birthdays and patterns drawn on the entry slip are chosen far more often than the underlying mathematics would suggest, which means those selections win a smaller share when they win at all. Choosing unpopular numbers does not improve the odds of winning. It improves the size of the prize if you do.

How draws became public institutions

Lotteries are among the oldest organised games of chance, and they are almost alone in having been used repeatedly as instruments of public finance rather than merely tolerated as entertainment. The mechanism is old: a fixed number of tickets sold, a public draw, a prize schedule known in advance, and a portion of the proceeds retained for the promoter's purpose.

That structure was used across early modern Europe to fund harbours, bridges, waterworks and building projects at a time when direct taxation was politically difficult and public borrowing was undeveloped. A lottery converted a large number of small voluntary payments into a single usable fund, and it did so without a tax collector. Where states did not run them, private and charitable schemes filled the space, often with far weaker guarantees that prizes would be paid at all.

That weakness drove the first wave of lottery regulation. Draws are easy to promise and easy to falsify, and the history of private lotteries includes enough failure to make the case for state monopoly or licensing on its own. The long-run pattern across many jurisdictions is a cycle: proliferation, scandal, prohibition, and eventual reintroduction under public control with audited draws and earmarked proceeds. The modern draw game, with its published odds and supervised equipment, is the settled end of that cycle rather than a new invention.