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Probability and odds

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

Every game of chance rests on a small number of measurable quantities. A probability is one of them: a number between zero and one that says how large a share of all equally likely outcomes belongs to the event you are interested in. Odds are the same information written as a ratio rather than a fraction, and the gap between the odds a game pays and the odds the outcome actually carries is where the whole commercial structure of chance games is built.

Counting outcomes

The classical definition of probability is a counting exercise. Write down every outcome a mechanism can produce, check that each one is equally likely, then divide the number of outcomes you care about by the total. A single die has six faces, so the probability of a four is 1/6. A single-zero roulette wheel has thirty-seven pockets, so any named pocket is 1/37. A shuffled deck has fifty-two cards, so a named card is 1/52.

The requirement that outcomes be equally likely is doing quiet work. It is true of an honest die and an honest wheel because the mechanism is symmetric. It is not automatically true of a slot machine reel, where different symbols can occupy different numbers of stop positions, and it is not true of a horse race, where the outcomes are not interchangeable at all. Where symmetry fails, probability has to be assigned by a model or by the design of the machine rather than by counting faces.

Writing the same number three ways

The same probability appears in three notations. Fractional odds state the ratio of losing outcomes to winning outcomes: a die roll of four is 5/1 against, because five of the six outcomes lose. Decimal odds state the total return for a unit stake including the stake itself, so the same roll is 6.00. Percentages state the probability directly, so the roll is 16.67%.

Converting between them is arithmetic, not judgement. If fractional odds are written a/b against, the implied probability is b / (a + b). If decimal odds are d, the implied probability is 1 / d. Running the conversion in both directions is the fastest way to see what a quoted price is actually claiming about the world.

True odds and paid odds

True odds describe the mechanism. Paid odds describe the contract. On a single-zero wheel a straight-up number wins once in thirty-seven, so the true odds against are 36/1. The payout offered is 35/1. Nothing about the wheel has been altered; the shortfall is entirely in the contract written over it.

That one-unit gap is the entire mechanism of the house edge. It is why a game can be perfectly honest, perfectly random and perfectly transparent, and still return less than it takes. The mechanism is not deception. It is a price.

Combining events

Two rules cover most of the combinations that arise in games. If events cannot both happen, their probabilities add: a roll of four or five is 1/6 + 1/6 = 1/3. If events are independent, their probabilities multiply: two fours in a row is 1/6 x 1/6 = 1/36.

Independence is the assumption that fails most often. Dice and wheels have no memory, so successive results are genuinely independent. Cards dealt from a deck without replacement are not: removing an ace changes the composition of what remains, which is why card games behave differently from dice games and why conditional probability matters there.

A tree diagram, as above, is the reliable way to keep multi-stage problems straight. Each path is a multiplication, each set of paths that ends in the same result is an addition, and the complete set of leaves must sum to one. If it does not, an outcome has been missed.

Why intuition fails

Human intuition about chance is poor in specific, predictable ways. People underestimate how often coincidences occur in large samples, overestimate the significance of short streaks, and treat independent events as though they were self-correcting. The last of these is common enough to have its own name, and it survives because a run of results genuinely does look patterned even when it was produced by a mechanism with no memory at all.

The correction is not better instinct but arithmetic. Count the outcomes, write down the probability, convert it to the notation the game uses, and compare it with what the game pays. Everything else in this site is an application of that one procedure.