Example 3.1.5.
Let \(X\) indicate an event \(A\) which has probability \(p\text{.}\) Then:
\begin{gather*}
\E(X) = 0 \cdot (1-p) + 1 \cdot p = p.
\end{gather*}
That is, the expected value of an indicator random variable is the probability of the event that it indicates.