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Exactas--How To Calculate Fair Exacta Odds

Posted By: Dunbar
Date: 16 May 02, 2:00 pm

When considering the attractiveness of an exacta bet, a quinella bet, or a place or show bet, it is necessary to have a good estimate of a horse’s chance of coming in second. If you already have a good estimate of the true probabilities of each horse winning a race, then the Harville formula gives a reasonable estimate of the probability that a given horse will come in 2nd.

EXACTAS

Say you have 8 horses in a race and each of them has an equal chance to win. What is a fair exacta payoff? Well, there are 8*7 possible exactas, so a reasonable estimate of the probability of each exacta, when the horses are equal, is 1/56, which is 55-1. So, whatever method we use to calculate fair exacta odds should give an answer consistent with 55-1.

Let’s consider a specific example. If horse #8 comes in first, then what is the chance that horse #5 will come in 2nd? The way Harville looked at it, this is the same as asking “what is the chance of horse #5 winning a 7-horse race, having removed the actual winner?” And the way he calculated that probability was:

Prob horse Y comes in 2nd given horse Z comes in 1st:
= PY/(1-PZ),

where PY is the best estimate of Horse Y's chance to come in 1st, and PZ is the best estimate of Horse Z's chance to come in 1st.

That is, he adjusts the odds of the (2nd place) horse to reflect the fact that we are looking at the case where another horse has finished first. In the example I started with, where each horse has an equal 1/8 chance to win, the Harville formula gives this probability for any horse coming in 2nd, given that some other horse has come in first:

=(1/8)/(1-(1/8)) = (1/8)/(7/8) = 1/7.

So, if Horse #8 comes in first with probability (1/8), and Horse #5 comes in 2nd with probability 1/7 (from Harville's formula), the exacta probability is (1/8) * (1/7) = 1/56, which is precisely what we know must be true.

Harville’s formula works just as well when horses have different probabilities to win. Just as an exercise, let me do a hypo on the Preakness. Let’s assume the following fair odds. (These are not necessarily accurate!)

Medaglia d’ Oro 5-2 (2/7)
War Emblem 4-1 (1/5)
Harlan’s Holiday 6-1 (1/7)

Assuming those odds are fair, what would be a fair payoff for various exactas:

1. War Emblem/Medaglia d’ Oro:
The chance of War Emblem coming in first is 1/5.
The chance of Medaglia d’ Oro coming in 2nd, given that War Emblem comes in first is:
= (2/7)/(1-(1/5)) = (2/7)/(4/5) = 0.36.
The chance that War Emblem comes in first (p=1/5) AND Medaglia d’ Oro comes in 2nd (p=0.36) is:
(1/5) * 0.36 = 0.071.
To get the odds, divide 1/0.071 = 14 for 1. So, a fair payoff on a $2 exacta bet on War Emblem over Medaglia d’ Oro would be $28.

How about the other way?:
2. Medaglia d’ Oro/War Emblem:
The chance of Medaglia d’ Oro coming in first is 2/7
The chance of War Emblem coming in 2nd, given that Medaglia is first is:
= (1/5)/(1-(2/7)) = 0.28.
The chance that Medaglia comes in 1st AND War Emblem comes in 2nd is:
(2/7)*0.28 = 0.08.
To get the exacta odds, divide 1/0.08 = 12.5 for 1. So, a fair payoff on a $2 exacta bet on Medaglia d’ Oro over War Emblem would be $25.

You can do the same for Harlan’s Holiday and any other horse in the race. In fact, it is easy to make up a table of fair exacta odds using the Harville Formula. Then, once you have established the fair win odds of both horses in an exacta, all you have to do is check the table to find the fair exacta payoff.

In the table below, the odds at the top are the fair win odds for the first horse in the exacta; the odds on the left are the fair win odds for the 2nd horse in the exacta. The numbers in the table represent the fair price for a $2 exacta based on those win odds.

FAIR EXACTA ODDS FOR VARIOUS FAIR WIN ODDS


Odds 1-1 3-2 2-1 5-2 3-1 4-1 5-1
1-1 4.00 6.00 8.00 10.00 12.00 16.00 20.00
3-2 5.00 7.50 10.00 12.50 15.00 20.00 25.00
2-1 6.00 9.00 12.00 15.00 18.00 24.00 30.00
5-2 7.00 10.50 14.00 17.50 21.00 28.00 35.00
3-1 8.00 12.00 16.00 20.00 24.00 32.00 40.00
4-1 10.00 15.00 20.00 25.00 30.00 40.00 50.00
5-1 12.00 18.00 24.00 30.00 36.00 48.00 60.00
6-1 14.00 21.00 28.00 35.00 42.00 56.00 70.00
7-1 16.00 24.00 32.00 40.00 48.00 64.00 80.00
8-1 18.00 27.00 36.00 45.00 54.00 72.00 90.00
9-1 20.00 30.00 40.00 50.00 60.00 80.00 100.00
10-1 22.00 33.00 44.00 55.00 66.00 88.00 110.00

--Dunbar










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Messages In This Thread

Exactas--How To Calculate Fair Exacta Odds -- Dunbar -- 16 May 02, 2:00 pm
Great Post, Dunbar! -- Jesse F. Knight -- 16 May 02, 4:06 pm
Absolutely- superb post-- -- D -- 16 May 02, 4:48 pm
I told this to a horse winner, and he told me .... -- ricky -- 16 May 02, 5:13 pm
completely agree -- Dunbar -- 16 May 02, 5:58 pm
Articles by Harville, Ziemba, Benter, etc -- 666 -- 17 May 02, 3:04 am
Ziemba Book -- Jesse F. Knight -- 17 May 02, 2:53 pm
Good luck -- StevieY -- 17 May 02, 3:22 pm
I have one -- ricky -- 17 May 02, 3:30 pm
No, you should sell it to me:) (nt) -- StevieY -- 17 May 02, 3:35 pm
auction -- ricky -- 17 May 02, 3:53 pm
if anyone cares, its out on ebay (nt) -- ricky -- 22 May 02, 2:26 pm
I ordered one from Amazon.com today -- Dunbar -- 17 May 02, 7:47 pm
I tried that twice... -- StevieY -- 17 May 02, 7:50 pm

 


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