Well let's break it down.

Basic probability states that...
P(A<N) = N/100
P(B<N) = N/100
P(A<N and B<N) = P(A<N)*P(B<N) = (N/100)^2
That's all easy. But that's not what we want to know.

In order for the average of A and B to be less than N, A+B must less than 2N. (again, simple)
So we can safely say P(A+B<2N) is the probability that the average of the two is less than 2N.

In order for A+B<2N to be true, A must be less than 2N and B must be less than 2N-A

In that case, basic probabiliy also states that...
P(A<2N) = 2N/100
P(B<2N-A) = (2N-A)/100
P(A<2N and B<2N-A) = (2N/100)*((2N-A)/100) = (4N^2-2NA)/1000

And there's your answer. Unless there's an error in my math, the probability that the average of A and B is less than N is (4N^2-2NA)/1000.


Edit: Just to clarify, A is the larger of the two numbers.

Nooooooooooo problem!

Glad I could help.

Thanks I always try my best to explain things thoroughly. Makes it easier when you don't have to keep on going back to things later.

Truthfully, no. I never took a stat class (though my Alg II teacher was also the APSTAT teacher)...
We did have a small unit on basic probability in that class too, but I figured this out mostly from logic. That's all probability is, anyway. I just looked at it and said, "What conditions must be true in order for this to be true?" and then, "What other conditions must be true in order for each condition to be true?" Using that thought process, I broke it down until it was at a point that was simple enough to understand and find an equation for.