Working Out Betting Odds
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.
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/
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