Apparently there is something called the Harville Formula for these classes of calculations.

I haven't bothered delving into that as I'd already derived the formula from first principles as follows:

The Trifecta probability is

abc/(1-a)/(1-a-b)

where a, b, and c are the win probabilities for runners A, B, C.


For every permutation instance A, B and C in your field calculate that probability and add it to slots A, B and C of an array.

When complete that array will have the required probabilities.