「Odds in statistics and Odds in a horse race」の版間の差分
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(同じ利用者による、間の17版が非表示) | |||
9行目: | 9行目: | ||
===A race that 5 horses start and 10 bettors bet=== | ===A race that 5 horses start and 10 bettors bet=== | ||
− | Here is a horse race that 5 horses start and | + | 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. | Names of horses are A, B, C, D and E. | ||
17行目: | 17行目: | ||
!style="width:50px"|Horses | !style="width:50px"|Horses | ||
|- | |- | ||
− | |A | + | |style="text-align:center"|A |
|- | |- | ||
− | |B | + | |style="text-align:center"|B |
|- | |- | ||
− | |C | + | |style="text-align:center"|C |
|- | |- | ||
− | |D | + | |style="text-align:center"|D |
|- | |- | ||
− | |E | + | |style="text-align:center"|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: | ||
+ | |||
+ | {|class="wikitable" | ||
+ | |- | ||
+ | !style="width:50px"|Horses | ||
+ | !style="width:80px"|Bettors | ||
+ | |- | ||
+ | |style="text-align:center"|A | ||
+ | |10 bettors | ||
+ | |- | ||
+ | |style="text-align:center"|B | ||
+ | |5 bettors | ||
+ | |- | ||
+ | |style="text-align:center"|C | ||
+ | |3 bettors | ||
+ | |- | ||
+ | |style="text-align:center"|D | ||
+ | |2 bettors | ||
+ | |- | ||
+ | |style="text-align:center"|E | ||
+ | |1 bettor | ||
+ | |} | ||
+ | |||
+ | ===Results=== | ||
+ | If each horse wins, pooled $20 are redistributed as following: | ||
+ | |||
+ | {|class="wikitable" | ||
+ | |- | ||
+ | !style="width:50px"|Horses | ||
+ | !style="width:80px"|Bettors | ||
+ | !style="width:140px"|If the horse wins | ||
+ | !style="width:130px"|Calculation | ||
+ | |- | ||
+ | |style="text-align:center"|A | ||
+ | |10 bettors | ||
+ | |$20 into 10 bettors | ||
+ | |$20 ÷ 10 = $2 | ||
+ | |- | ||
+ | |style="text-align:center"|B | ||
+ | |5 bettors | ||
+ | |$20 into 5 bettors | ||
+ | |$20 ÷ 5 = $4 | ||
+ | |- | ||
+ | |style="text-align:center"|C | ||
+ | |3 bettors | ||
+ | |$20 into 3 bettors | ||
+ | |$20 ÷ 3 = $6.66.. | ||
+ | |- | ||
+ | |style="text-align:center"|D | ||
+ | |2 bettors | ||
+ | |$20 into 2 bettors | ||
+ | |$20 ÷ 2 = $10 | ||
+ | |- | ||
+ | |style="text-align:center"|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: | ||
+ | |||
+ | {|class="wikitable" | ||
+ | |- | ||
+ | !style="width:50px"|Horses | ||
+ | !style="width:80px"|Bettors | ||
+ | !style="width:140px"|If the horse wins | ||
+ | !style="width:130px"|Calculation | ||
+ | !style="width:80px"|Winning probability | ||
+ | |- | ||
+ | |style="text-align:center"|A | ||
+ | |10 bettors | ||
+ | |$20 into 10 bettors | ||
+ | |$20 ÷ 10 = $2 | ||
+ | |10/20 | ||
+ | |- | ||
+ | |style="text-align:center"|B | ||
+ | |5 bettors | ||
+ | |$20 into 5 bettors | ||
+ | |$20 ÷ 5 = $4 | ||
+ | |5/20 | ||
+ | |- | ||
+ | |style="text-align:center"|C | ||
+ | |3 bettors | ||
+ | |$20 into 3 bettors | ||
+ | |$20 ÷ 3 = $6.66.. | ||
+ | |3/20 | ||
+ | |- | ||
+ | |style="text-align:center"|D | ||
+ | |2 bettors | ||
+ | |$20 into 2 bettors | ||
+ | |$20 ÷ 2 = $10 | ||
+ | |2/20 | ||
+ | |- | ||
+ | |style="text-align:center"|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: | ||
+ | |||
+ | <nowiki>*</nowiki>''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)''''' | ||
+ | |||
+ | {|class="wikitable" | ||
+ | |- | ||
+ | !style="width:50px"|Horses | ||
+ | !style="width:80px"|Bettors | ||
+ | !style="width:140px"|If the horse wins | ||
+ | !style="width:130px"|Calculation | ||
+ | !style="width:80px"|Winning probability | ||
+ | !style="width:100px"|Odds (reduced) | ||
+ | |- | ||
+ | |style="text-align:center"|A | ||
+ | |10 bettors | ||
+ | |$20 into 10 bettors | ||
+ | |$20 ÷ 10 = $2 | ||
+ | |10/20 | ||
+ | |10:10 (1:1) | ||
+ | |- | ||
+ | |style="text-align:center"|B | ||
+ | |5 bettors | ||
+ | |$20 into 5 bettors | ||
+ | |$20 ÷ 5 = $4 | ||
+ | |5/20 | ||
+ | |15:5 (3:1) | ||
+ | |- | ||
+ | |style="text-align:center"|C | ||
+ | |3 bettors | ||
+ | |$20 into 3 bettors | ||
+ | |$20 ÷ 3 = $6.66.. | ||
+ | |3/20 | ||
+ | |17:3 | ||
+ | |- | ||
+ | |style="text-align:center"|D | ||
+ | |2 bettors | ||
+ | |$20 into 2 bettors | ||
+ | |$20 ÷ 2 = $10 | ||
+ | |2/20 | ||
+ | |18:2 (9:1) | ||
+ | |- | ||
+ | |style="text-align:center"|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: | ||
+ | {{quote|content= | ||
+ | If you bet $1 to a horse with odds of '''9:1 (9-1)''' and the horse wins,<br>you will earn '''$9''' (usually called "payout" in a real horse race)<br>and additionally '''$1 back''' that you bet;<br>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'''. |
2022年11月7日 (月) 00:25時点における最新版
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.