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 9:1 (9-1) and the horse wins, |
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.
