Probability – Calculate Probability of a Specific Horse Finishing in nth Place

conditional probabilityprobabilitystatistics

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.

UPDATE

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

or

sum of HorseXWin * HorseY2nd * (HorseZWin/(sum of remaining win probabilities)), for each X/Y

or

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.

Best Answer

First of all lets assume that if horseD has twice the chances of horseF to finish 1st in a race of 7 horses, then it also has twice the chances to finish first when there are 100 horses or 2 horses or n horses. (It is not defined whether this is true or not. I'll continue assuming this.)

Let's take your example: "calculate the chance of HorseC finishing in 2nd place"

If horseA finishes 1st (0.35), then the probability of horseC finishing 2nd (1st amongst the others) is $\frac{0.15}{0.25+0.15+0.10+0.09+0.05+0.01}=\frac{0.15}{1-0.35}$. So this probability is $0.35\frac{0.15}{1-0.35}$.

Likewise if horseB finishes 1st, then the probability of horseC finishing 2nd is $0.25\frac{0.15}{1-0.25}=P_B\frac{P_C}{1-P_B}$.

The total probability of horse i to finish 2nd is the sum of these:

$P(i,2)=\sum_{i\neq j}{P_j\frac{P_i}{1-P_j}}$.

Can you continue in order to find a general formula for $P(i,n)$?

Related Question