I break down horse races into two horse races and my probabilities are fairly accurate. I have been wrestling with how to convert all these probabilities into an accurate estimate of each horse winning the race. An actual sample of a race with a five horse field is below. In this sample race I show all possible pairs of horses and their probabilities. In line 1 Horse #1 is running against Horse #2 and has a probability of .462 of beating Horse #2, Horse #1 agianst horse #3 has a probability of .375, etc.
How do I figure the true odds of all the horses winning the race based on these two horse matchups?
I've looked at multiplying the win probs for each horse and then assigning probability based on their % of the total. For example, horse 1 is .462 x .375 x .52 x .465 = .0419, horse 2, etc. Once I have the totals for each horse I add them up and then calculate the percentage of each. In the example below, #3 has a 45% chance to win (.1538/.3435), # 1 has a 12% chance, etc. Does this sound right? Any other approaches would be appreciated.
I guess it is not sound to claim that "Horse 1 beat Horse 2" and "Horse 1 beat Horse 3" are independent. Since Horse 1 winning the first place is equivalent to Horse 1 beat all other horses, i.e. the intersection of all these events, and you cannot simply multiplying them together. You may lack the joint structure to determine the joint probability.
BMG, thanks for your response. You are correct that my probabilities predict the win horse will beat all others. Similarly, the horse predicted to run 2nd beats all the remaining horses and so forth. Based on your comment it looks like you are not recommending multiplying the probabilities and joint probability may not be an option. Any further thoughts? I admit I am not a probability expert but I do know a bit about horse racing.
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