Concept
What is the formula for the Law of Total Probability?
Stephen Davies, Ph.D. Version 2.2.2 Through Discrete Mathematics A Cool Brisk Walk / Chapter 1
"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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