State the law of total probability.

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Answer

$B_{1}$
$Used constantly: decompose a complex probability by conditioning on a partitioning variable (age, disease status, region).$
$Foundation of many probabilistic derivations, mixture models, and marginalization in Bayesian inference.$

Check yourself — multiple choice

$P(A)\; = \;P(A \mid B)$
For a partition { $B_{i}$ }: P(A) = Σ $P(A \mid B_{i})$ · $P(B_{i})$ — decompose by conditioning on partition
P(A) = 1 - P(A)
Only for independent events

Total probability: sum conditional probs weighted by prior over a partition.

#probability#theory

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