# How does $\Bbb P(A) = \Bbb E(I_A)$ translate in plain English?

$\Bbb P(A) = \Bbb E(I_A)$

If we translate this identity in plain English,

Probability of an event $A$ is equal to the Expectation of the indicator function $I_A$?

What does it actually mean?

$$\mathbb{I_A}=\left\{ \begin{array}{ll} 1, & x \in A\\ 0, & x \not \in A\\ \end{array} \right.$$
$$E(\mathbb{I_A}) = 1\times P(X \in A) + 0\times P(X \not \in A) = P(X \in A)$$
Therefore the expected value of the indicator function for an event $A$ is the same as the probability of $A$ happening.