I have been interested in calculating a specific horse's chance of finishing in nth place given every horse's chance of winning in a particular race.
i.e. Given the following:
Horse Chance of winning A 0.35 B 0.25 C 0.15 D 0.10 E 0.09 F 0.05 G 0.01
Calculate for any horse it's chance of finishing in nth place. For instance, calculate the chance of HorseC finishing in 2nd place, or calculate the chance of HorseB finishing in 3rd place.
I thought that this is something I could get help with online. All of the journal articles I found discuss chances of a pair of horses winning, i.e. the chance of horse A winning and horse B coming second, which is obviously different to this question.
This: http://forum.sbrforum.com/handicapper-think-tank/526381-win-v-place-odds-value-math-question.html#post5076725 is the closest thing I have found to what I am looking for, but I believe he assumes that given one horse wins, the others have an equal chance of placing second, which is clearly not the case.
I just had time to have a go at this and I am trying to come up with a formula for P(i, n) as @ThanosDarkadakis suggested.
I am unsure of whether the odds of HorseZ finishing 3rd is:
sum of HorseXWin * HorseY2nd * (HorseZWin/(1-HorseXWin-HorseY2nd)), for each X/Y
sum of HorseXWin * HorseY2nd * (HorseZWin/(sum of remaining win probabilities)), for each X/Y
sum of HorseXWin * HorseYWin * (HorseZWin/(1-HorseXWin-HorseYWin)), for each X/Y
Where HorseX is the winner of the race, HorseY comes 2nd and HorseZ comes 3rd (for each X/Y).
I'm sure that with a formula for P(i, 3) it would be trivial to write a formula for P(i, n). Any suggestions are greatly appreciated.