Odds

In probability theory and statistics, odds are a way of expressing probabilities. Mathematically, odds are calculated as follows

Where the value of the chance and the probability that the event occurs. The function is called the odds function (or odds function). The chance is therefore the quotient of the probability that an event occurs and the probability that it does not occur (counter probability).

One speaks of a 1 in 1 chance that heads will appear on a coin toss or a 1 in 5 chance that a 6 will appear on a dice roll.

If the value of a chance is one, then this is identical to a 50:50 chance. Values greater than one express that the probability in the numerator has the greater value, while values less than one mean that the one in the denominator is greater.

If you know the probabilities, you know the chances and vice versa,

so that the introduction of chances seems in a way superfluous. But also in probability theory there are problems in the solution of which chances play a more important and natural role than probabilities themselves, as for example in the judicial evaluation of circumstantial evidence, see Bayesian inference, or in the odds strategy for the calculation of optimal decision strategies.

In statistics, the so-called odds ratio is used to evaluate the difference between two odds and thus to make statements about the strength of relationships. With a chance ratio, however, the clear relationship between chances and probabilities is lost.

Bets

In the context of betting, especially sports betting, the English term odds is often translated as betting, win or win rate or odds for short. Odds have long been the usual way for bookmakers to indicate probabilities. This is also the origin of the name of the German sports bet Oddset. The representation of odds in the betting business varies depending on the location (see also article odds)

Example 1

If we consider an event with the probability of occurrence of 1 out of 5 (i.e. 0.2 or 20%), then the odds are 0.2/(1-0.2) = 0.2/0.8 = 0.25. If 0.25 is staked in a fair bet and the event occurs, the profit is 1; if 1 is staked, the profit is thus 4, plus the stake of 1 is returned. A bookmaker in continental Europe gives 5.0 for this. The stake of 1 to be paid back is already included in the payout here, this is also called the gross odds. A British bookmaker writes 4 to 1 against (or 4/1), because the net profit is only four times the stake, British bookmakers always give the net odds (mostly in fractional representation), an American bookmaker gives with +400 the profit from a stake of 100.

Example 2

If, on the other hand, the probability of an event occurring is 4 out of 5 (i.e. 0.8 or 80%), then the odds are 0.8/(1-0.8) = 4. If 4 is staked in a fair bet and the event occurs, the winnings are 1, plus the stake of 4 is paid back. A bookmaker in continental Europe gives 1.25 for this, the stake is already included in the payout here. A British bookmaker writes 4 to 1 for (or 1/4), an American bookmaker specifies with -400 the necessary stake to achieve 100 profit.

In the above calculation, it is assumed that the distribution of bets corresponds to the actual probabilities. In reality, however, the bookmaker rather tries to predict the betting behavior, because if he predicts it correctly, he will in any case collect the predetermined bookmaker margin and thus avoid unnecessary risk. Therefore, instead of using the probability of an event, he uses the probable bets on that event to calculate the odds.