「Odds in statistics and Odds in a horse race」の版間の差分

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===Converting winning probabilities into odds===
 
===Converting winning probabilities into odds===
−
Here winning probabilities are converted into odds as follows:
+
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]'''''
+
<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"
 
{|class="wikitable"
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!style="width:130px"|Calculation
 
!style="width:130px"|Calculation
 
!style="width:80px"|Winning probability
 
!style="width:80px"|Winning probability
−
!style="width:80px"|Odds
+
!style="width:100px"|Odds (reduced)
 
|-
 
|-
 
|style="text-align:center"|A
 
|style="text-align:center"|A
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|$20 &divide; 10 = $2
 
|$20 &divide; 10 = $2
 
|10/20
 
|10/20
−
|10:10
+
|10:10 (1:1)
 
|-
 
|-
 
|style="text-align:center"|B
 
|style="text-align:center"|B
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|$20 &divide; 5 = $4
 
|$20 &divide; 5 = $4
 
|5/20
 
|5/20
−
|15:5
+
|15:5 (3:1)
 
|-
 
|-
 
|style="text-align:center"|C
 
|style="text-align:center"|C
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|$20 &divide; 2 = $10
 
|$20 &divide; 2 = $10
 
|2/20
 
|2/20
−
|18:2
+
|18:2 (9:1)
 
|-
 
|-
 
|style="text-align:center"|E
 
|style="text-align:center"|E
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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:
 
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 the horse 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've got '''$10''' in your pockets}}
+
{{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.
 
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

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,
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.

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