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Stats question - predicting winners from normal distributions

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Help me out here... I have 5 independent normal distributions X1 thru X5 with differing means and stddev. I'd like to compute the probability that a random variable from X1 is greater than a random variable picked from X2 thru X5.

Here's how far I've gotten. I basically have to figure out the P(X1 > X2) and P(X1 > X3) and P(X1 > X4) and P(X1 > X5). I can compute those independently fairly easily. P(X1 > X2) is the CDF of a normal at 0 with mean equal to the difference of means and stddev equal to the square of the sum of the squares of the stddevs. Rinse and repeat for the other 3 probabilities.

So I have those 4 probabilities that X1 beats the other 4 but then I'm stuck. The distributions are no longer independent (they all depend on X1) so I can't just multiply the probabilities together. What's the next step? My initial thought was just multiply them together and scale them so the sum of the probabilities equals one and that's a "good enough" answer but if I can get this exactly right, that would be super.

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