Conditional Probability

Why the direction matters

A medical test’s sensitivity P(positive|disease) is not the same as the chance of disease after a positive result P(disease|positive). The second depends on how common the disease is; Bayes’ theorem connects them. Confusing these conditionals is a common reasoning error.

For a sequence of events, the chain rule multiplies successive conditional probabilities. The events need not be independent; each factor accounts for what is already known.

A two-way table makes the denominator visible

Suppose a class has 12 students who ride the bus, and 4 of them arrive late. There are 30 students total, with 10 late. Then P(late | bus)=4/12=1/3, while P(bus | late)=4/10=0.4. Both conditionals use the same overlap—the four students who are both bus riders and late—but the condition determines the denominator. Read the vertical bar as “given that” and identify the group after it before dividing.

Conditioning changes the reference group. In a deck, P(king)=4/52, but P(king | face card)=4/12. Independence is precisely the case where conditioning on B leaves A's probability unchanged. If P(B)=0, the elementary ratio definition is undefined; more advanced models need a separate definition. For continuous variables, conditioning on an exact value with probability zero also requires a density or limiting argument. Use Bayes’ theorem when the evidence is known but the hypothesis is the quantity you need to infer.

Try this above

Select an evidence group in the visualiser and compare the event’s probability before and after conditioning. Check that the restricted denominator matches the selected group.

For three events, the chain rule gives P(A∩B∩C)=P(A)P(B|A)P(C|A∩B). This is true without assuming independence; each factor updates the probability using what has already occurred. It is useful for sequential draws, where the chance changes after each draw. For example, drawing two aces without replacement is (4/52)(3/51), not (4/52)². The squared expression would model independent draws with replacement. Always check whether the process returns an item or changes the next trial.

The multiplication rule is often easier than the conditional formula when outcomes occur in sequence. If the chance of rain is 0.2 and the chance of a traffic delay given rain is 0.5, then P(rain and delay)=0.2×0.5=0.1. Reversing the condition asks a different question and needs Bayes' theorem. A probability tree or contingency table helps expose the path and its denominator, especially when several conditions are involved.

Sources and next steps

• OpenStax: Introductory Statistics 2e covers conditional probability and multiplication rules.
• See Bayes’ theorem for reversing the condition using base rates.