Set of Elementary Events belonging to k Events is Event

Theorem
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A_1, A_2, \ldots, A_m$ be events in the event space $\Sigma$ of $\EE$.

Let $S$ denote the set of all elementary events of $\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \ldots, A_m$.

Then $S$ is an event of $\Sigma$.

Proof
Let $r_1, r_2, \ldots r_k$ be a set of $k$ elements of $\set {1, 2, \ldots, m}$.

Then:
 * $\paren {A_{r_1} \cap A_{r_2} \cap \cdots \cap A_{r_k} } \setminus \paren { A_{r_{k + 1} } \cup A_{r_{k + 2} } \cup \cdots \cup A_{r_m} }$

contains exactly those elements of $\Omega$ which are contained in exactly those events $A_{r_1}, A_{r_2}, \ldots, \cap A_{r_k}$.

By Elementary Properties of Event Space, that last set is an event of $\EE$.

By selecting all combinations of $\set {1, 2, \ldots, m}$, we can form the subsets of $\Sigma$ containing all those elementary events of exactly $k$ events.

Each one is an element of $\Sigma$.

We then form the union $S_k$ of each of those element of $\Sigma$.

By Elementary Properties of Event Space, $S_k$ is an event of $\EE$.

By construction, $S_k$ contains all and only the elementary events of $\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \ldots, A_m$.

Also see

 * Elementary Properties of Event Space