Suppose we have the random walk $S_t=\sum\limits_{i=1}^t X_i$, where $$X_i = \begin{cases} -1 & \text{with probability } p \\ 0 & \text{with probability } q \\ 1 & \text{with probability } 1-p-q\end{cases}$$

Define $\tau$ as the time at which $S_n$ hits one for the first time, i.e. $\tau=\min\{t>0 : S_t=1\}$.

Take $n\in \mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$ with $k<n$. Conditional on $\tau=n$, what is the probability that the number of time periods $i<n$ in which $X_i=0$ is equal to $k$?, i.e. how do you compute $$\mathbb{P}\left( \left|\{i\in\{1,\dots,n\}:X_i=0\}\right|=k \right)\ ?$$


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