Law of Total Probability

Probability · Axiom Academy

LESSON Law of Total Probability Computing probabilities by partitioning the sample space 1. Partitioning the Sample Space A partition divides the sample space S into disjoint events B₁, B₂, ..., Bₙ that cover all possible outcomes. Each outcome belongs to exactly one partition element. Any event A can be broken down according to the partition. Event A occurs with B₁, or with B₂, or with B₃, etc. These are mutually exclusive pathways. 3. The Law of Total Probability Using the multiplication rule on each piece, we express P(A) as a weighted sum of conditional probabilities, weighted by the probabilities of the partition elements. This law is especially useful when we know conditional probabilities P(A|Bᵢ) and partition probabilities P(Bᵢ), but don't directly know P(A). Common in medical testing and quality control.

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