Assume you're tossing a fair coin; you win only if you get (i+1) consecutive heads right after (i) consecutive tails; what is the probability this game never stops?
My attempt is as below: $P$(never stop)$=$$P$(1 T before 2H|1T)$*$$P$(1 T before 3H|1T)$*...=\prod_{n=1}^{\infty}(1-\frac{1}{2^{n+1}})$
However, could anyone verify my solution? Also, does $P$(never stop) converges to 0?
