# Probability that a point from one normal distribution is greater than points taken from several other distributions?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that each is greater than all the others.

For example: I have random variables $A$,$B$,$C$, and $D$. I want to know:

$$P(A>Max\{B,C,D\})$$ $$P(B>Max\{A,C,D\})$$ $$P(C>Max\{A,B,D\})$$ $$P(D>Max\{A,B,C\})$$

I have been trying to research it myself; however, I am not getting to a good answer. I think that this page would have something to do with the solution, but I don't know how to extend it to multiple distributions: Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

My assumption was that I could find the probability that one was greater than or equal to each of the others and multiply them together. However, when I do this for all distributions the total is around $93\%$. Meaning that there is a $7\%$ that none are the highest. This doesn't make sense to me, especially because I was calculating greater than or equal.

I have been attempting this for a while and keep getting incorrect results.

Any help would be greatly appreciated.

• Are you doing this for general mean and standard deviations for each variable or do you have specific parameters to work from? – CommonerG May 5 '13 at 22:54
• This may have been answered before. See if this helps. math.stackexchange.com/questions/232535/… – CommonerG May 5 '13 at 23:08