[math-fun] Sampling from 2 different Gaussians

I'm pretty rusty with my probabilities. Suppose that I have 2 different Gaussian distributions, each with its own mean & variance. I want to take one sample from each of the distributions & compare them. Suppose that we call the first distribution A and the second distribution B; let a be the sample from A and b be the sample from B. Is there a relatively simple closed form expression for p(a>b) in terms of Amean, Avar, Bmean, Bvar ? What about a simple closed form for asymptotics -- e.g., a,b are sampled from a tail of A,B respectively -- e.g., a,b are each in the top decile of their respective distribution? top 1% ? I'm particularly interested in the cases where (Amean=Bmean,Avar>Bvar) and (Avar=Bvar,Amean>Bmean).