The problem is this:

We have a number N of runners each known to cover the distance at a mean time Ti (i=1 to N) and the standard deviation around the mean time is the same, say ó.
What are the probablitities of runner i winning the race ?


Typical cases:

100 meters sprint: mean values are of the order of 10'' and ó is of the order of 0.1''

1000 meters horse race: mean values are of the order of 60'' and ó is of the order of 0.5''


This can be computed fairly easily with the use of normal pseudo-random numbers.
But I am looking for a closed solution or approximation.