So, for the longest time, I had it in my head that when rolling 2d20 and taking the better result (2d20b1), the average result was 15. I don't know where I got this number from (probably from extrapolating from the far more trivial case of 2d2b1, a pair of coins), but suffice it to say, it's wrong. Today I learned of my mistake and a better estimator.
As it turns out, a good estimator is that for rolling a d20 twice and taking the better result, the average result is actually 13.825. A good estimation for die-rolling (and the actual exact average result if working with a random number generator not restricted to whole numbers only
) is that the average = (min + 2*max) / 3, or that average = min + (max - min)*2/3.
Now, I haven't double-checked my math, but I think that the generalized case of XdNb1 (rolling X N-sided dice and taking the best result) has an average of approximately (1 + X*N) / (X+1). Again, dice only give integer results, so this is only an estimate; with a random number generator that could give any result from A to B, not just whole numbers, the average would be exactly (A + X*B) / (X+1), assuming my napkin math is right. If anybody feels like double-checking me, I'd appreciate it.