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The race is over 5 furlongs (1000 meters) and for each of the six contestants it is known that their probable times at this distance are:

horse 1: 57.00 sec

horse 2: 57.20 sec

horse 3: 57.35 sec

horse 4: 57.80 sec

horse 5: 58.10 sec

horse 6: 59.50 sec

But, as is always the case in horse races, these times are uncertain, so the outcome is unknown.

In fact each of the above times is accurate by plus or minus 0.50 seconds, i.e. for horse "1" there is a Gaussian distribution with mean 57.00 and standard deviation 0.5, for horse "2" there is a Gaussian with mean 57.20 and st. dev. 0.5 and so on.

What is the probability for each horse to win the race ?

There is an easy (but a little slow) answer that can be derived by Monte Carlo simulation using random numbers, but it's not what I 'm asking for.

Does anyone know a functional approximation for the winner's pdf ?