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### Working Out Betting Odds

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Date: 06/27/2001 at 08:48:59
From: Phil Baggaley
Subject: Working out betting odds

I am doing a horse racing night in a local pub, and as part of the
committee I have been asked to try to find out how to work out the
betting odds for each horse. Any money that we collect is to go
straight back out into prize money.

Your help would be greatly appreciated.
```

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Date: 06/28/2001 at 18:40:50
From: Doctor Jaffee
Subject: Re: Working out betting odds

Hi Phil,

If you know the probability of winning for each horse, you can convert
that fraction into odds, or vice-versa.

For example, suppose there are 5 horses (A,B,C,D,and E) in a race, and
their respective probabilities of winning are:

80   10   5    3        2
---, ---, ---, ---, and ---.
100  100  100  100      100

Notice that the total is 100/100.

Theoretically, if these horses were to race 100 times, horse A would
win 80 and lose 20, so the odds of winning are 80 to 20 or 4 to 1.

Likewise, the odds for the next four horses are 1 to 9, 1 to 19, 3 to
97, and 1 to 49.

To determine the payoffs, then, if the probability of winning is x/y,
you should pay the bettor y dollars for each x dollars bet. For
example, if I bet 80 dollars on horse A, I would collect 100 dollars
at the booth. That includes my original 80 dollars with 20 dollars
profit.

Next, we need to calculate the payoffs for second place finishes.
If horse A wins, the odds of horse B beating the others is 10 to
(5 + 3 + 2), or 10 to 10, which is equivalent to the probability
10/20 = .5  So, the probability of A winning and B coming in second is
.8 x .5 = .4

Likewise, the probability of C winning and B coming in second is
.05(10/95) = .0052632

The probability of D winning and B coming in second is .03(10/97) =
.0030928
The probability of E winning and B coming in second is .02(10/98) =
.0020408

So, the probability of B coming in second is the sum of these
probabilities, which is .4103968

Also, there is a probability of 0.1 that B will win, so the
probability of a bettor being successful on a wager that B will place
is .4103968 + 0.1 = .5103968 or 5103968/10000000.

If you divide the top and bottom of this fraction by 5103968, you get
approximately 1.96. In other words, you should pay \$1.96 to an
individual who bets \$1 to place on horse B.

Determining the payoffs for show is more complicated. You would have
to compute the probabilities that a horse comes in third with respect
to two other horses coming in first and second.

I hope this helps. Write back if any of my explanation requires
clarification or if you have any other questions. Good luck with your
night at the races.

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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