Thank you for writing this out, I didn't quite get what was going on at first. But then, to formalize the recursion from your example: let's assume we're at item n in the iterator, and at that point we've selected a winner from the previous n-1 items with equal probability, i.e. each item had a 1/(n-1) chance of being selected. The probability that item n will override it is 1/n. The probability that the old winner will remain selected is thus (n-1)/n. That means that the old winner remains selected with probability 1/(n-1) * (n-1)/n, which cancels out to 1/n, so each item is indeed selected with equal probability in the end.