# Odds in statistics and Odds in a horse race

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## Definition of Odds in statistics

$\displaystyle{ Odds=\frac{probability\ that\ the\ outcome\ occurs}{probability\ that\ the\ outcome\ does\ not\ occur} }$

## Example of a horse rase

### Assumption

In order to simplify, the bookmaker in the following scenario does not charge any betting fees

### A race that 5 horses start and 10 bettors bet

Here is a horse race that 5 horses start and 20 people — 20 bettors — bet each of 5 horses.

Names of horses are A, B, C, D and E.

Horses
A
B
C
D
E

Each bettor bets $1 for one horse. All bet money are pooled ($20 pooled in total) and will be redistributed for bettors who bet a winning horse.

Now each all bettors bet as following:

Horses Bettors
A 10 bettors
B 5 bettors
C 3 bettors
D 2 bettors
E 1 bettor

### Results

If each horse wins, pooled $20 are redistributed as following: Horses Bettors If the horse wins Calculation A 10 bettors$20 into 10 bettors $20 ÷ 10 =$2
B 5 bettors $20 into 5 bettors$20 ÷ 5 = $4 C 3 bettors$20 into 3 bettors $20 ÷ 3 =$6.66..
D 2 bettors $20 into 2 bettors$20 ÷ 2 = $10 E 1 bettor$20 into 1 bettor $20 ÷ 1 =$20

### From a view point of winning probabilities

Assumed that predictions by bettors were as precise as God, a winning probability of each horse is as follows:

Horses Bettors If the horse wins Calculation Winning probability
A 10 bettors $20 into 10 bettors$20 ÷ 10 = $2 10/20 B 5 bettors$20 into 5 bettors $20 ÷ 5 =$4 5/20
C 3 bettors $20 into 3 bettors$20 ÷ 3 = $6.66.. 3/20 D 2 bettors$20 into 2 bettors $20 ÷ 2 =$10 2/20
E 1 bettor $20 into 1 bettor$20 ÷ 1 = $20 1/20 ### Converting winning probabilities into odds Here winning probabilities are mathematically converted into odds as follows: *note that here odds are [probability the horse does not win]:[probability the horse wins], or we can rephrase as odds against winning (odds for losing) Horses Bettors If the horse wins Calculation Winning probability Odds (reduced) A 10 bettors$20 into 10 bettors $20 ÷ 10 =$2 10/20 10:10 (1:1)
B 5 bettors $20 into 5 bettors$20 ÷ 5 = $4 5/20 15:5 (3:1) C 3 bettors$20 into 3 bettors $20 ÷ 3 =$6.66.. 3/20 17:3
D 2 bettors $20 into 2 bettors$20 ÷ 2 = $10 2/20 18:2 (9:1) E 1 bettor$20 into 1 bettor $20 ÷ 1 =$20 1/20 19:1

### Interpretation of Odds in a horse race

Here we've got odds in a horse race mathematically.

Generally speaking, if odds are given as "18:2" or "9:1" (usually written as "9-1" in a real horse race) against a horse, it means as follows:

 If you bet $1 to a horse with odds of 9:1 (9-1) and the horse wins,you will earn$9 (usually called "payout" in a real horse race)and additionally $1 back that you bet;Now you have$10 in your pockets

See the horse D in the previous table.

The odds against horse D were 18:2, i.e., 9:1 (9-1).

A bettor on D bet $1 and got$10 as a result of redistribution of pooled $20, which means the bettor earned$9 by betting \$1 on horse D.

Here mathematically calculated odds are completely same as odds commonly shown in a horse race.