Reading odds and probability

A price on a game of chance is a promise about what a stake returns if a particular outcome occurs. The three notations in common use say the same thing in different arithmetic. Fractional notation, written 5/1, states the profit against the stake: five units won for every one risked, with the stake returned alongside. Decimal notation, written 6.00, states the total return including the stake, so it is always the fractional figure plus one. Moneyline notation states either the profit on a hundred-unit stake for an unlikely outcome, written +500, or the stake required to profit a hundred units on a likely one, written -125.

Nothing is lost or gained in the translation between them. They exist because different betting cultures grew up around different arithmetic conveniences: fractions suited a spoken ring where prices were called aloud, decimals suit a screen where returns are multiplied, and the moneyline suits a market where most outcomes sit close to even. A reader who can move between all three can compare any two prices without being distracted by the format.

Every price implies a probability. For decimal odds the implied probability is simply one divided by the decimal figure: a price of 4.00 implies 0.25, or twenty-five per cent. For fractional odds of a/b the implied probability is b divided by the sum of a and b, so 3/1 implies one in four, the same twenty-five per cent. This conversion is the single most useful piece of arithmetic in the whole subject, because it turns an unfamiliar quoted number into a statement about how often something is being said to happen.

The word implied is doing real work. The implied probability is not a claim about the world; it is a claim about the price. It tells you what chance would make that price break even, and nothing more. The true chance of an outcome in a designed game—a die, a wheel, a deck, a draw machine—can be computed exactly from the mechanism. The gap between the true chance and the implied chance of the price offered against it is the entire subject of the next section of this board.

Take any complete set of mutually exclusive outcomes and convert each quoted price to an implied probability. The true probabilities of a complete set must sum to exactly one, because something must happen. The implied probabilities of a real quoted set never do. They sum to more than one, and the excess is the margin. On a two-outcome market where both sides are genuinely even, true probabilities are 0.50 and 0.50; a market quoting 1.91 on each side implies 0.5236 twice over, a total of 1.0471, and that 4.71 per cent excess is the structural cost of participating.

This is not concealment. The arithmetic is available to anyone willing to do the division, and the sum is the plainest single diagnostic available for any priced game: add up the implied probabilities and see how far past one hundred per cent the book runs. A reader who does this habitually stops thinking of prices as offers and starts reading them as a distribution with a deliberate surplus built into it.

Ordinary speech uses the word odds loosely, and the looseness causes real confusion. Odds against an outcome compare failures to successes: 4/1 against means four ways to lose for every one way to win, a probability of one in five. Odds on an outcome reverse the comparison: 1/4 on means the outcome is expected four times as often as not, a probability of four in five. Probability itself compares successes to the total, not to the failures, which is why one in five and four to one against describe the same event.

The distinction matters because the two conventions are routinely mixed in casual writing about chance, and a factor-of-one-step error in either direction changes a fair price into a poor one. Whenever a figure about a game of chance is quoted without stating which convention it uses, the safe response is to convert it to a plain probability and work from there.