Law of total probability/expectation

The probability of an event can be written as a weighted sum of conditional probabilities.

The expected value of a random variable can be written as a weighted sum of conditional expected values.

If {Aᵢ}ᵢ is a finite or countably infinite partition of the sample space, then

P(B) = ∑ᵢ P(B | Aᵢ) P(Aᵢ)

E(X) = ∑ᵢ E(X | Aᵢ) P(Aᵢ)

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