what is the basis for the development of the following product

Question

I have the following identity in my probability text book, but I was unable to relate it to a known development. Here, $A_n$ is a series of events.

Answer 1

If you expand the product, there are $2^n$ terms. The first one arises from taking the $1$ in each binomial factor $1 - 1_{A_k}$, yielding $1^n=1$. Each of the other $2^n-1$ terms corresponds to a nonempty subset $S \subseteq \{1,2,\dots,n\}$. Note that the product of characteristic functions is the characteristic function of the intersection: $1_{A} 1_{B} = 1_{A \cap B}$.

Putting it all together: \begin{align} \prod_{j=1}^n \left(1 - 1_{A_j}\right) &= \sum_{S \subseteq \{1,\dots,n\}} \prod_{s \in S} \left(-1_{A_s}\right) \\ &= \sum_{S \subseteq \{1,\dots,n\}} (-1)^{|S|} \prod_{s \in S} 1_{A_s} \\ &= \sum_{S \subseteq \{1,\dots,n\}} (-1)^{|S|} 1_{A_{\prod_{s \in S}}} \\ &= \sum_{k=0}^n \sum_{\substack{S \subseteq \{1,\dots,n\}:\\ |S|=k}} (-1)^k 1_{A_{\prod_{s \in S}}} \\ &= 1 + \sum_{k=1}^n (-1)^k \sum_{\substack{S \subseteq \{1,\dots,n\}:\\ |S|=k}} 1_{A_{\prod_{s \in S}}} \\ &= 1 - \sum_{k=1}^n (-1)^{k-1} \sum_{\substack{S \subseteq \{1,\dots,n\}:\\ |S|=k}} 1_{A_{\prod_{s \in S}}} \\ \end{align}

This is a generalization of the binomial theorem $$\prod_{j=1}^n (1-x) = (1-x)^n = \sum_{k=0}^n \binom{n}{k} (-1)^k x^k.$$