Expected value and probability of stopping time

Question

I have a task:

Take $(X_n)$ i.i.d. $P(X_k=1)=P(X_k=-1)=\frac{1}{2}$. $$t=\inf\big\{n: X_1+X_2+...+X_{n-1}+X_n=2015\big\}.$$ I proved that $t$ is stopping time.

I have to prove that: $P(t<\infty)=1$ and I have to calculate $Et$.

I tried from Wald's equation, but $EX_1=0$.

Thanks in advance.

Answer 1

Set $S_n=X_1+...+X_n$. Then, $S_n\sim\mathrm{Binom}(n,1/2)$. Let $$\tau=\inf\{n\mid S_n=2015\}.$$ Remark that $$\mathbb P\{\tau=k\}=\begin{cases}0&k\leq 2014\\\binom{k-1}{2014}\frac{1}{2^{k}}&k\geq 2015\end{cases}.$$ I let you conclude.