Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E \in \mathcal{F}$ is an D1716: Event in $P$
(ii)	$I_E : \Omega \to \mathbb{R}$ is an D2796: Indicator random boolean number on $P$ with respect to $E$

Let $n \in \{ 1, 2, 3, \ldots \}$ be a D5094: Positive integer.

Using result R2482: Expectation of discrete random variable, we have \begin{equation} \begin{split} \mathbb{E} I^n_E & = 1^n \cdot \mathbb{P}(I_E = 1) + 0^n \cdot \mathbb{P}(I_E = 0) \\ & = 1 \cdot \mathbb{P}(I_E = 1) + 0 \cdot \mathbb{P}(I_E = 0) \\ & = \mathbb{P}(I_E = 1) \\ \end{split} \end{equation} $\square$