Odds in statistics and Odds in a horse race

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 converted into odds as follows:

*note that here odds are [probability the horse does not win]:[probability the horse wins]

Horses	Bettors	If the horse wins	Calculation	Winning probability	Odds
A	10 bettors	$20 into 10 bettors	$20 ÷ 10 = $2	10/20	10:10
B	5 bettors	$20 into 5 bettors	$20 ÷ 5 = $4	5/20	15:5
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
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 the horse and the horse wins, you will earn $9 (usually called "payout" in a real horse race) and additionally $1 back that you bet