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For a given horse race, I have the predicted distributions of each horse running the race like below,

enter image description here

Image taken from http://www.cms.zju.edu.cn/UploadFiles/AttachFiles/20054191750380.ppt

I want to calculate what is the probability of each horse winning the race? based on the above normal distributions. And after estimating the same I want the probabilities to add up to 1.

The x-axis in the diagram above is the running time of horses. Hence the horse with the lowest time (left-most) could be a potential winner.

I know the formula that whenever we have two independent normal distributions, then P(X1 <= X2) i.e. the probability of point taken from distribution X1 will be less than or equal to point taken from distribution X2 can be given as,

ss.norm.cdf(-(mu1 - mu2) / np.sqrt(sigma1 + sigma2))

Proof:

import numpy as np
from scipy import stats as ss

mu1, mu2, sigma1, sigma2 = 0., 100., 1., 1.
X2 = np.random.normal(mu2, sigma1, size=(10000,))
assert(np.mean([np.mean(np.where(np.random.normal(mu1, sigma1) <= X2 , 1, 0)) for _ in range(10000)])) == ss.norm.cdf(-(mu1 - mu2) / np.sqrt(sigma1 + sigma2))

So how do I extend the above code to find the winning probabilities of every horse, and that they should add up to 1?

Milind Dalvi
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    This is a minor variation of https://stats.stackexchange.com/questions/358181. The answer is the same: you need to perform numerical integration. – whuber Jan 15 '21 at 21:07
  • Adding another reference https://stats.stackexchange.com/questions/74091/px-1-minx-i-ldots-x-n-across-different-normal-random-variables – Milind Dalvi Jan 15 '21 at 21:37
  • Thank you for finding that duplicate. I have posted an answer with working R code to illustrate the numerical integration solution. It is far more accurate and efficient than any simulation and works with any number of "horses." – whuber Jan 15 '21 at 22:06

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