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Suppose we are given a simple random walk starting in $0$, i.e. $(X_k)_{k\in\mathbb{N}}$ with $P[X_k=+1]=P[X_k=-1]=\frac{1}{2}$. What is the probability of hitting the level $a$ before hitting the level $b$, where we assume $b<0<a$ and $|a|\le |b|$. Let's define

$$T_a:=\inf\{n|S_n=a\}$$ and similarly $$ T_b:=\inf\{n|S_n=b\}$$ where $S_n:=\sum_{i=1}^nX_i$. Therefore we are interested in the probability $$P[T_a< T_b]$$ I think one needs the reflection principle at some point, but I'm not sure how it is exactly applied.

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Hint: Since $(S_n)_{n \in \mathbb N}$ is a martingale, you can use the fact that $$\begin{align*}E[S_T]&=aP(S_T=a)-bP(S_T=b)=\\&=aP(S_T=a)-b(1-P(S_T=a))=(a+b)(P(S_T=a))-b=\\&=(a+b)P(T_a<T_b)-b\end{align*}$$ (where $S_T=\inf\{n|S_n \in \{a,b\}\}$) and the optional stopping theorem, i.e. that $$E[S_T]=E[S_0]=0$$ to deduce that $$P(S_T=a)=P(T_a<T_b)=\frac{b}{a+b}$$

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