Law of Total Probability

Formula

\[ P(B) = \sum_i P(B\mid A_i)\,P(A_i) \]

Parameters

• \(\{A_i\}\): partition of the sample space
• \(P(A_i)>0\) and \(\sum_i P(A_i)=1\)

What it means

Computes \(P(B)\) by conditioning on a partition of cases.

What it's used for

• Decomposing probabilities by cases.
• Mixture models and diagnostic calculations.

Key properties

• Extends to integrals for continuous partitions
• Basis for Bayes' rule normalization

Common gotchas

• The \(A_i\) must be mutually exclusive and exhaustive.
• Don't forget to include all cases in the partition.

Example

If \(P(B)=0.3\), \(P(A\mid B)=0.8\), \(P(A\mid eg B)=0.2\), then \(P(A)=0.8\cdot0.3+0.2\cdot0.7=0.38\).

How to Compute (Pseudocode)

Input: event probabilities / joint distribution entries
Output: requested probability quantity

identify the relevant events/variables and required joint/marginal terms
apply the probability identity in the card formula
check denominator/normalization terms are valid (nonzero when required)
return the computed probability

Complexity

• Time: Typically \(O(1)\) for a single event computation once required probabilities are available; larger table-based calculations scale with table size
• Space: \(O(1)\) extra space for a single computation
• Assumptions: Probability terms (joint/marginals/conditionals) are already known or computed separately

See also

• Bayes' Rule