A friend of mine enjoys horse racing and is a regular small time bettor. He has roughly broken even over several decades of betting. I do not bet, but I am interested in the processes he uses to determine his bets.
Along those lines I came up with an idea to possibly increase his handicapping success. I asked him, for every race, to divide the horses into three groups, group Y included horses which had a chance of winning the race. Group M were those horses that could, maybe, win. And group N were horses he thought had no chance of winning. Of course, some races were won by horses in groups M and N.
After some time we computed the ratio of horses in group Y which went on the win the race. That ratio is W/Y where Y is the total number of horses ranked Y, and W is the number of those that did win. Some, even most, races have more than one Y candidates. It turns out the ratio of W/Y was 3/8 for my friend.
What I was hoping to achieve is a measure of what odds a Y rated horse should pay before that horse would be bet. It isn't enough to reckon that a horse is the best choice to win, the odds that horse pays must be high enough to overcome the uncertainty.
I have stumbled on to a conceptual error in my thinking. In some ways, W/Y does represent a probability but I am unable to rationalize that interpretation given that some horse must win every race, and that the a-priori probabilities for the various horses must add up to 1. Given that most races have more than one Y candidate, their chances of winning are not independently related to W/Y. Clearly W/Y does not represent the probability that a Y candidate will win.
Everything becomes simpler if I interpret W/Y as the chance (probability) that in a given race with at least one Y rated horse, that a(any) Y rated horse will win. Given that interpretation I can say that if there are m Y rated horses in a given race then the probility that one of them will win is, on average, W/Y. Given that, and assuming that all Y rated horses are equivalent, then each Y rated horse has a probability of W/(m*Y) of winning. To be a good bet, then, the Y rated horse must pay better than m*Y/W.