Law of total probability

The law of total probability expresses an event’s probability as the sum of its conditional probabilities across the mutually exclusive, exhaustive cases of a partition.

Probability partition: A collection of mutually exclusive events whose union is the entire sample space. The theorem requires these cases to be mutually exclusive and exhaustive.
Sample space: The set of all possible outcomes in a probability model. The partition’s cases must together cover this entire set.
Diagnostic testing: The use of tests to infer whether a condition or property is present. Overall test outcomes combine conditional rates across people with and without the condition.
Inclusion–exclusion principle: A formula for the probability of a union that corrects for overlaps among events. Unlike a partition sum, it handles events that may overlap.
Multiplication rule for probability: A rule stating that a joint probability equals a conditional probability multiplied by the conditioning event’s probability. It supplies the weighted terms that the theorem sums.
Mutually exclusive events: Events that cannot occur together. This condition prevents outcomes from being counted in more than one case.
Base rate fallacy: A reasoning error that neglects prior frequencies when evaluating conditional evidence. The theorem makes the base-rate weighting behind diagnostic probabilities explicit.
Bayes' theorem: A theorem that reverses conditional probabilities using prior probabilities and a marginal probability. It uses total probability to reverse conditioning rather than merely decompose an event.
Law of total expectation: A theorem expressing an expectation as the expected conditional expectation given a random variable. It extends the same averaging-over-cases structure from probabilities to expectations.
Exhaustive events: Events whose union contains every possible outcome. Without exhaustive cases, the sum could omit part of the event’s probability.

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