A fair coin is flipped $2n$ times. If the number of "heads" and the number of "tails" coincide, a tie is reached.
What is the probability $p_k$, that the number of ties occuring is exactly $k$, where $k=0,1,...,n$ ?
I tried to model this situation with a random walk. Then the number of ties is equal to the number of "roots" of the random walk. It is easy to calculate the probability that a tie is achieved after a specified number of steps, but the events "a tie is reached after $2,4,6,...$ steps" are dependent.