At the risk of reopening an inflammatory thread ... Quoting James Propp <jamespropp@gmail.com>:
A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2
Given the large collection of probability "paradoxes": Any sensible person would prefer a relatively straightforward calculation with easily understandable components, over any result based on meta-level reasoning, with its possible hidden pitfalls. For the mechanics, multiplying the series by 1 - 1/2 converts it to a familiar sum. This does require accepting that the series manipulation is safe, based on absolute convergence. (1 - 1/2) * (1/2 + 2/4 + 3/8 + ...) = (1/2 - 1/4) + (2/4 - 2/8) + (3/8 - 3/16) + ... = 1/2 + (2/4 - 1/4) + (3/8 - 2/8) + (4/16 - 3/16) + ... = 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, so the original series must sum to 2. Rich