You're playing a game. You start with a box with one black marble
and one white marble, and you sample twice with replacement. If
you select the white marble both times, you win. If you select the
black marble either time, you add another black marble and try
again. On each round, you sample twice with replacement, winning
if you select the white marble twice, otherwise adding another
black marble and moving on to the next round.
What is the probability that you eventually win? Equivalently, if
P(n) is the probability that you win on or before round n, what is
the limit of P(n) as n -> infinity?
The odds of winning the first time are 1/2² -- the odds of getting the white ball are 1/2, and you need to manage it twice; thus, P(1)=1/2². The odds of winning the second time are 1/3², so P(2)=1/2²+1/3². In general, you find P(n)=1/2²+1/3²+...+1/(n+1)². The limit is ζ(2)-1 (ζ() is the Riemann function) about 64.5%.