Total Probability Formula

Topics: Partition of a Sample Space - Probability - Sample Space

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(theorem)

Let $A\in \mathcal{F}$ (any event in the sample space; sigma-algebra) and let $(B_i{)}_{i\in \mathbb{N}}$ be a partition of the sample space. Then:

$$\begin{array}{rl}\mathbb{P}(A)& =\sum _{i\in \mathbb{N}}\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)\\ & =\sum _{i\in \mathbb{N}}\mathbb{P}(A\cap B_i)\end{array}$$

(see conditional probability and the third axiom for a probability measure)

We call this the total probability formula.