Two players, $A$ and $B$, alternately and independently flip a coin and the first player to obtain a head wins. Player $A$ flips first. What is the probability that $A$ wins?
Official answer: $2/3$, but I cannot arrive at it.
Thought process: Find the probability that a head comes on the $n$th trial. That's easy to do, it's just a Geometric random variable. Then find the probability that $n$th turn is player's A turn. Finally, multiply both probabilities.
When I came up with each probability, both of them depended on the amount of trials $n$, so my answer was a non-constant function of $n$.
However, what I find quite fantastic is that the answer is a constant, so the amount of tries until a head comes doesn't seem to matter.