There are several answers already given for working out the probability of one random variable being greater than another, but I can't make the leap to working out the probability of one random variable being greater than several others. My random variables are independent and normally distributed. For example:
Let $X$, $Y$ and $Z$ be independent normal random variables. What is $P(X>Y,X>Z)$?
The obvious (to me) answer, being to just multiply the two probabilities $P(X>Y)$ and $P(X>Z)$ does not work because the difference random variables $(X-Y)$ and $(X-Z)$ are not independent.
Edit
For $P(X>Y)$ the answer is: $$ {\rm P}(X > Y ) = \Phi \left(\frac{\mu_X - \mu_Y }{\sqrt{\sigma_Y^2 + \sigma_Y^2}}\right). $$ I'm hoping for a way of adding a third normal random variable to the equation. If this is possible I presume the answer can be easily expanded to add further random variables.