# Chance (stochastics)

In probability theory and statistics, a chance ( English odds ) is a way of specifying probabilities. For example, there is a 1: 1 chance that heads will appear on a coin toss or a 1: 5 chance that a 6 will appear when the dice is rolled. Mathematically, chances are calculated as quotients from the probability that an event will occur and the probability that it will not occur ( counter-probability ):

${\ displaystyle R (A) = {P (A) \ over {1-P (A)}}}$

Where is the value of the chance and the probability that the event will occur. If the value of a chance is one, then that is the same as 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 probability in the denominator is greater. ${\ displaystyle R}$${\ displaystyle P (A)}$${\ displaystyle A}$

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

${\ displaystyle P (A) = {R (A) \ over {1 + R (A)}} \ ,,}$

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

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

## Bets

In connection with betting , in particular with sports betting , which is English term opportunities often with competition, Victory or profit share or short- ratio translated. Odds have long been the standard way for bookmakers to quote probabilities. The name of the German sports betting Oddset is derived from this. The presentation of the chances in the betting business varies depending on the location (see also article profit ratio )

example 1
If you consider an event with a probability of occurrence of 1 out of 5 (i.e. 0.2 or 20%), then the chances 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; with a bet of 1 the win is 4, and the bet of 1 is paid back. A bookmaker in continental Europe gives it 5.0. The bet of 1 to be repaid is already included in the payout; this is also called the gross odds . A British bookmaker writes 4 to 1 against (or 4/1), since the net profit is only four times the stake, British bookmakers generally give the net odds (mostly in fractions), an American bookmaker gives the profit from one with +400 Bet from 100 on.
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 you bet 4 in a fair bet and the event occurs the profit 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. A British bookmaker writes 4 to 1 for (or 1/4), an American bookmaker states the stake necessary to make a profit of 100 at −400.

The above calculation assumes that the distribution of the stakes corresponds to the actual probabilities. In reality, however , the bookmaker tries to predict betting behavior because, if he predicts it correctly, he will in any case collect the bookmaker's margin set in advance and thus avoid unnecessary risk. Instead of the probability of an event, he therefore uses the likely bet stakes on that event to calculate the odds.

## literature

• Dennis V. Lindley: Understanding Uncertainty , Wiley, 2006, ISBN 978-0-470-04383-7 , limited preview in Google Book Search.
• F. Thomas Bruss : The art of making the right decision. In: Spectrum of Science . 06/2005. Spektrum der Wissenschaft Verlagsgesellschaft, pp. 78–84, , ( online ).
• Fahrmeir, Ludwig; Artist, Rita; Pigeot, iris; Tutz, Gerhard: Statistics. The way to data analysis . Heidelberg u. a .: Springer, 4th edition 2003, chapter 3.2.1.

## Individual evidence

1. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , and Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 114.