Probability and expected value of uniform random variables sum

Question

Let $X_1, \dots , X_n$ be i.i.d. random variables with $X_i \sim U_{(0,1)}$. Let $S_n= \displaystyle \sum\limits_{k=1}^n X_k$ and $T=\mathrm{inf}_n\{n: S_n > 1\}.$

Show: $P(T>n)=\frac{1}{n!}$ (1) and $\mathbb{E}(S_T)=\frac{e}{2}$ (2).

Our attempt: We were able to show $\mathbb{E}(T)=e$ using (1). But we don't get to show (1) and (2).

Answer 1

Hint

1) $$\mathbb P\{T>n\}=\mathbb P\{S_n\leq 1\}.$$

2) Notice that $\mathbb E[S_T]$ is a r.v. depending on $T$. In particular, $$\mathbb E[S_T]=\sum_{k=1}^T \mathbb E[X_i].$$