[math-fun] how many records?

Here's a cute and useful fact. Let S be a set of N numbers. Permute them into random order. Scan thru the elements in that order. Each time you encounter a new record max, ring a bell. So for example, if S={7*,8*,5,3,4,9*} you would ring the bell 3 times at the *s. EASY CLAIM: expected number of times the bell rings, equals HarmonicNumber(N) = 1+1/2+1/3+...+1/N. HARDER QUESTIONS, which I have not looked at: What about the variance? What about tail bounds? Can you write a formula for the whole probability distribution for the number of bell-rings: Prob(R) = xxx?