I found this in my math book. I have the answer but no explanation for it. I don't have any ideas on how to explain the formula.
$N$ persons are playing for $2m$ coins the following way: First, person 1 throws all the coins. If exactly half of them are heads, they win the game and all coins. If not the turn goes to person 2 who does exactly the same thing. If none of the $N$ persons win the first round, the turn goes back to person 1, and the game continues. The probability that person $k$ wins is $\frac{r(1-r)^{k-1}}{1-(1-r)^N}$ where $r=\frac{\binom{2m}{m}}{2^{2m}}$. Can you explain the formula?