How games of chance are designed
Par sheets, symbol weighting, the separation of return from hit frequency, what certification of a selection mechanism establishes, and what it does not.
The specification comes first
A modern game of chance is specified before it is built. The specification, usually called a par sheet, lists every symbol or outcome, the weight attached to each, every prize in the schedule, and the return that the combination of weights and prizes produces. The return is not measured after the fact; it is computed from the specification, and the finished game is then tested to confirm it behaves as specified.
Weights, not appearances
The central mechanism is that the visible symbols on a reel need not be equally likely. A mapping assigns each visible symbol a number of internal positions, and the selection is made uniformly across those internal positions rather than across the symbols. A symbol occupying one internal position in sixty-four appears once in sixty-four selections, whatever its size on the display and however many times it is printed.
Visible symbols on one reel: 12 Internal positions mapped: 64 top symbol -> 1 position -> 1/64 = 0.0156 second symbol -> 2 positions -> 2/64 = 0.0313 low symbols -> 61 positions between them Nothing on the display indicates which mapping applies.
This is why the appearance of a game says nothing about its probabilities, and why the near-miss - a high-value symbol landing just outside the paying position - carries no information. Positions adjacent to a paying position can be, and often are, weighted differently from the paying position itself.
Return and hit frequency are set separately
Two properties of a game are commonly conflated: how much it returns, and how often it returns anything at all. They are independent design choices, and the same return can be delivered in very different ways. Take a thousand equally likely outcomes and one unit staked per trial.
Design A Design B
200 outcomes pay 1 60 outcomes pay 5
50 outcomes pay 5 15 outcomes pay 20
10 outcomes pay 20 5 outcomes pay 50
2 outcomes pay 100
returned per 1,000 trials returned per 1,000 trials
200 x 1 = 200 60 x 5 = 300
50 x 5 = 250 15 x 20 = 300
10 x 20 = 200 5 x 50 = 250
2 x 100 = 200
------- -------
850 850
return = 850/1000 = 85 % return = 850/1000 = 85 %
hits = 262/1000 = 26.2 % hits = 80/1000 = 8.0 %
Both designs return the same proportion per unit staked. One returns something on roughly one trial in four and the other on one in twelve, so their results feel entirely different while their arithmetic is identical. Volatility is exactly this choice, and it is why two games advertising the same return can produce completely different distributions of outcome.
The selection mechanism
The outcome itself comes from a random number generator. In practice this is a deterministic algorithm whose output passes statistical tests for uniformity and independence and whose internal state cannot be inferred from the outputs. Certification of such a mechanism tests three separate things: that outcomes are distributed as the specification says, that successive outcomes are independent, and that the state cannot be predicted or reproduced from observed results.
Certification concerns the mechanism, not the prize schedule. A generator can be perfectly uniform and fully certified while the game it drives retains fifteen per cent of amounts staked, because the retention lives in the prizes rather than in the selection. Uniform selection and a negative expected value are entirely compatible, and in commercial games they always coexist.
Progressive prizes
A progressive prize is funded by diverting a fixed share of each amount staked into a pool that grows until the qualifying result occurs. The diverted share is part of the game's return, but it is delivered to one ticket at an unknown future point rather than spread across trials. The stated return of such a game is therefore an average over the whole life of the pool, and the return experienced during any particular stretch, when the qualifying result does not occur, is lower than the stated figure by the size of the diverted share.
Rules as arithmetic
Small rule variations move the arithmetic more than they appear to. Whether a tie returns the stake or loses it, whether a particular result pays at the full schedule or a reduced one, whether an extra unmatched outcome is present - each changes the count of ways to win or the size of a prize, and therefore the expected value. Comparing games by their prizes without also comparing their rules produces the wrong answer, which is the same lesson the two wheels give.