A credit model that ignores a downgrade, a hedge that ignores an earnings announcement, a fraud detector that ignores a flagged transaction — every one of these is a probability calculation that failed to update on available information. Conditional probability is the formal mechanism for that update: given that an event has occurred, what is the new probability of ? This note builds the definition and shows it is itself a legitimate probability law (Proposition 1), develops the multiplication rule for building up probabilities of multi-stage experiments (the tool behind every probability tree), and derives the two workhorse results built from it: the total probability theorem (decompose an unknown probability over a partition of scenarios) and Bayes' rule (invert the direction of conditioning — the mathematics behind every "given the evidence, what caused it?" question, from radar detection to credit inference).
Learning objectives. By the end of this note you should be able to:
- define and verify it satisfies the three probability axioms as a law on its own;
- apply the multiplication rule to compute the probability of a multi-stage outcome by reading products off a probability tree;
- apply the total probability theorem to decompose an event's probability over a partition of scenarios;
- apply Bayes' rule to invert a conditional probability, and interpret the result as updating a prior belief on new evidence.
In this lesson
- Conditional probability
- The multiplication rule and probability trees
- The total probability theorem
- Bayes' rule
Before you start
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