Question 13
(FRG 3) A particle moves from $(0,0)$ to $(n,n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n,y)$, $y\lt n$, it stays there if a head comes up and at $(x,n)$, $x\lt n$, it stays there if a tail comes up. Let $k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n,n)$.