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

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?

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First let's define the indicator function:

$$ \mathbb{I_A}=\left\{ \begin{array}{ll} 1, & x \in A\\ 0, & x \not \in A\\ \end{array} \right. $$

Now, find the expectation:

$$E(\mathbb{I_A}) = 1\times P(X \in A) + 0\times P(X \not \in A) = P(X \in A)$$

Therefore the expected value of the indicator function for an event $A$ is the same as the probability of $A$ happening.

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