I came across this problem while studying game theory. In this case, $i$ is a dummy player of a TU-game, but I think this identity is independent of the choice of $i$.
Let $N=\{1,2...n\}$ and $i\in N$, show that $$\sum_{i\notin S\subseteq N}\frac{1}{n\binom{n-1}{|S|}} =1$$
I've tried some algebraic proofs but got nowhere, and with combinatorial proofs I didn't get to the answer either. I haven't really had any practice with combinatorial proofs dealing with subsets of this kind. I did find that the amount of subsets of $N$ that do not contain $i$ is equal to $2^{n-1}$, but I couldn't find out a way to say anything about the size of these subsets.
Hint : note that \begin{eqnarray*} \sum_{i\notin S\subseteq N}= \sum_{S\subseteq N-\{i\} } =\sum_{j=0}^{n-1} \sum_{ \stackrel{S\subseteq N-\{i\}}{ \text{and} \mid S \mid=j } } =\sum_{j=0}^{n-1} \binom{n-1}{j} . \end{eqnarray*}