I feel kind of stupid, but I've wasted enough time, so...
Say I have three Gaussian random variables, a,b,c. What is Pr(a>b,a>c)?
The two things are not independent, so Pr(a>b,a>c)=Pr(a>b)*Pr(a>c|a>b).
Now I'm stuck on the conditional. Thanks for any help.
The real problem I want to solve is if I have N gaussians, what is the probability of a given one being bigger than all the rest. But one step at a time.
Oops, I had jumped to the assumption that a, b, and c were independent and identically distributed variables. If they are not identically distributed then my previous remark does not apply, and the only way I know to find the answer to your question is to apply numerical methods to approximate the integral of the joint pdf of a, b, and c over the region where a is greater than b and c. I don't know how to evaluate the integral otherwise.