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What is the formula for the Law of Total Probability?

What is the formula for the Law of Total Probability? (Brisk · Chapter 1) — "In the two-set case, no matter what the event A is, we can divide up its probabili…

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Law of total probability
Two-set case
Event A is divided into its parts inside B and outside B
Pr(A) = Pr(A ∩ B) + Pr(A ∩ Bᶜ)
Using conditional probability, Pr(A) = Pr(A | B)Pr(B) + Pr(A | Bᶜ)Pr(Bᶜ)
B may be any other event
General case with a partition of Ω
The events Cₖ, for k from 1 to N, form a partition of Ω
Pr(A) = ∑ₖ₌₁ᴺ Pr(A | Cₖ)Pr(Cₖ)
The summation symbol Σ represents a cumulative sum
The counter k starts at 1, ends at N, and adds the expression for each value

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What is the formula for the Law of Total Probability?

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Brisk

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USR292600b83e2a

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"In the two-set case, no matter what the event A is, we can divide up its probability like this: Pr(A) = Pr(A ∩ B) + Pr(A ∩ Bᶜ) = Pr(A | B) Pr(B) + Pr(A | Bᶜ) Pr(Bᶜ), where B is any other event. The last step makes use of the conditional probability definition. We’re dividing up A into the B’s and the non-B’s, in a strategy to determine A’s probability. In the general case, if N sets named Cₖ, where k is a number from 1 to N, make up a partition of Ω, then: Pr(A) = Pr(A ∩ C₁) + Pr(A ∩ C₂) + ··· + Pr(A ∩ Cₙ) = Pr(A | C₁) Pr(C₁) + Pr(A | C₂) Pr(C₂) + ··· + Pr(A | Cₙ) Pr(Cₙ) = ∑ₖ₌₁ᴺ Pr(A | Cₖ) Pr(Cₖ). The symbol Σ represents a cumulative sum: the counter k starts at 1 and goes up to N, adding the expression to the right of the sign each time."

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