# Odds in statistics and Odds in a horse race

## 目次

## Definition of Odds in statistics

[math]\displaystyle{ Odds=\frac{probability\ that\ the\ outcome\ occurs}{probability\ that\ the\ outcome\ does\ not\ occur} }[/math]

## 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 |

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**.