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give a simple symmetric random walk, denote $\mathbb P_0(T_1 < \infty)$ the probability that starting at $0$ and the first time reaching level 1 within finite discrete time steps. What is $\mathbb P_0(T_1 < \infty)$?

So can I assume that it hits level $1$ within $n$ steps, then use the hitting time thm to calculate the probability? Or is there another way to this?

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The probability you're asking is 1.

In fact, the probability of reaching any point within a finite number of steps is always 1.

An intuitive way to think about it is: " Any finite sequence will appear infinitely often"

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