Defining a problem with random variables.

I am studying the following problem.

Take n random normal distributions each with a common standard deviation of s. Call these d(1), d(2)...d(n).
Let the means of these distributions be m(1) m(2)...m(n).
Let r(n) be a random variable taken from d(n).
Let S be the set of r(1),r(2)...r(n). So each distribution contributes one variable.
Let p(x) be the probability that r(x) is the largest value in S.

I am interested in p(1)...p(n) and their relationship, if any, with m(1)...m(n) and s.

I am an amateur at mathematics and my usual way of researching problems is to look around for papers or books covering the area I am dealing with.

My problem is I have no idea what mathematicians/statisticians call this type of problem.

Can anyone tell me how it might be described so I can search for it or point me towards papers or texts where it may be covered?


Thanks in advance for your help.