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How games of chance are designed

Field: Mathematics. This page was last modified on 17 August 2026.

A game of chance is a specification before it is an entertainment. A designer chooses a random mechanism, enumerates its outcomes, attaches a prize to each, and then adjusts the schedule until three numbers land where they are wanted: the return, the frequency of winning, and the volatility. Everything visible in the finished game is arranged around that arithmetic.

The paytable is the game

The paytable is a list of outcomes with a prize beside each one. Multiply every prize by the probability of the outcome that earns it, add the results, and the total is the game's return per unit staked. Subtract that from one and the remainder is the operator's share. Everything else in the design, the artwork, the sound, the pacing, sits on top of a table that has already fixed the economics.

Because the return is a sum, a designer has many degrees of freedom. Raising a top prize can be paid for by trimming a mid-tier prize, by making a common outcome slightly rarer, or by removing a small prize entirely. Each of these keeps the return constant while changing how the game feels, and choosing between them is the substance of the work.

Hit frequency

Hit frequency is the proportion of plays that return anything at all. It is deliberately independent of return: a game can pay something on a third of plays and still return less than a game that pays on one play in ten, if the rarer game's prizes are larger in proportion.

Very low hit frequency makes a game feel unresponsive. Very high hit frequency creates a different problem, because many of the returns must then be smaller than the stake to keep the arithmetic in balance, and a sequence of prizes smaller than the amount staked is easy to misread as a sequence of wins. Both effects are known to designers and both are chosen, not accidental.

Volatility

Volatility, the standard deviation described in the variance article, is the third target. Low-volatility designs concentrate the return in frequent small prizes, so results stay near the expectation and a given stake lasts longer. High-volatility designs concentrate it in rare large prizes, so results wander a long way from the expectation in both directions.

The table below shows three schedules built on the same simple mechanism with the same return, differing only in how the return is distributed. The point of the comparison is that return alone describes a game very poorly.

Weighting the mechanism

Where a physical mechanism cannot supply the probabilities a designer wants, the mapping between the mechanism and the display supplies them instead. A reel with a modest number of visible positions can be driven by a much longer list of internal stops, with several internal stops pointing at a common symbol and only one pointing at a rare one. The visible strip looks even; the underlying distribution is not, and the difference is a design parameter. This technique is described in more detail in the article on slot machines.

The same principle appears without any machinery in games built on a draw from a large set. A lottery does not need weighting at all, because choosing six numbers from a large pool already produces probabilities small enough to support very large prizes; the design work there is in the prize tiers and the pooling rules rather than in the draw.

Verification

Because the specification is arithmetic, it can be checked. A regulator or an independent test house can recompute the return from the paytable, confirm that the outcome probabilities in the software match the declared ones, and run the random number generator through statistical tests for uniformity and independence. The regulation article traces how that verification became a normal part of the industry rather than an exception.