Probability, odds and the gap between them

Counting the outcomes

A game of chance is defined by two lists. The first is the set of things that can happen, which mathematicians call the sample space. The second says what each of those things pays. Almost every confusion about gambling arithmetic comes from working with only one of the two lists.

Start with the simplest possible device. A fair six-sided die has six outcomes, each equally likely, so the probability of any single face is one in six. The probability of an even number is three of the six faces, which is one half. The probability of a number greater than four is two of six, or one third. Nothing here requires anything beyond counting: list the favourable outcomes, divide by the total, and the fraction is the probability.

The counting stays easy as long as the outcomes are genuinely equally likely. When they are not, the same method still works but the list has to be built more carefully. Two dice thrown together do not produce eleven equally likely totals from two to twelve; they produce thirty-six equally likely ordered pairs. A total of seven arises from six of those pairs and a total of twelve from exactly one, which is why seven appears six times as often as twelve over a long series of throws.

Odds are a ratio, not a fraction

Odds express the same information as a comparison rather than a share. Where probability asks what fraction of all outcomes are favourable, odds ask how many unfavourable outcomes there are for each favourable one. The die that shows a six with probability one in six has odds of five to one against, because five faces fail for every one that succeeds.

Written in the British fractional style, a price of 5/1 means five units of profit for every one unit staked, returning six units in total when it wins. The decimal style used across much of Europe folds the stake back in and writes the same price as 6.00, meaning six units returned in total per unit staked. Neither notation contains more information than the other, and both convert to a probability by the same step.

The rule behind the third column is short. Divide one by the decimal price and the result is the probability that price implies. A decimal price of 6.00 implies one sixth. A fractional price of a/b implies b divided by the sum of a and b, so 5/1 implies one divided by six, the same answer reached by the other route.

Where the gap opens

Because the conversion is exact, a stated price is a statement about probability whether or not it was intended as one. That makes the arithmetic useful as a check. If a game has six equally likely outcomes and pays 4/1 on a correct call, the price implies a probability of one in five while the game supplies one in six. The gap between those two numbers is the entire commercial basis of the game, and it is discussed in the entry on house edge and expected value.

The same gap explains a common misreading. People often describe a price as an estimate of how likely something is, and for events with no fixed sample space, such as the result of a contest, that is roughly what a price is trying to be. But it is always an estimate with a margin built in, and the margin points in one direction. Treating an implied probability as a neutral forecast quietly ignores the part of the number that exists to be retained rather than paid out.

Independence and the gambler's fallacy

One further property of the die matters more than any other. The die has no memory. After four consecutive sixes the probability of a fifth six is still one in six, because the mechanism that produces the next throw is unchanged by the previous ones. Outcomes with this property are called independent, and most gambling devices are built to have it.

The belief that a run of one outcome makes the opposite outcome due is known as the gambler's fallacy, and it survives because it confuses two different true statements. It is true that long runs of one outcome are rare. It is also true that once a run has occurred, it is in the past and cannot influence the next throw. The rarity attaches to the sequence viewed in advance, not to the remaining throws viewed from the middle of it.

Card games dealt from a finite shoe are the genuine exception, and the exception is instructive precisely because it is so narrow. There the outcomes are not independent: every card dealt changes what remains, so the sample space shrinks as the deal proceeds. That is a property of physical cards being removed from a deck, and it disappears the moment the deck is reshuffled or the game is redesigned to draw from a fresh distribution each time.