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.