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Let's consider an asymmetric Random Walk on $Z$, with transition probabilities $p_{i, i+1}=p$, $~~p_{i, i+1}=q$, $\forall i \in \mathcal{Z}$, $p+q=1$ and $p>q$.

I am interested in the probability of the first hitting time of a bareer in $i=n$, assuming that the walk started from $i=0$ at time $t=0$. Is there an explicit formula for it? How does it depend on $n$?

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This is an exercise. The reflection principle lets you count the walks which first arrive at $n$ at time $t$. These all have the same probability, since they all have $n$ more positive steps than negative steps. – Douglas Zare Mar 11 '13 at 17:28
This has been answered fully in… – Jeremy Voltz Mar 11 '13 at 17:56
The answer by Did suggests another solution by generating functions but the reflection method is much simpler. – Douglas Zare Mar 11 '13 at 20:14

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