Which rule is associated with conditional probability?

The association of Bayes' theorem with conditional probability comes from its foundation in updating the probability of an event based on new evidence. Bayes' theorem mathematically expresses how the probability of a hypothesis (or event) changes when new information is available. It incorporates prior knowledge (the initial probability of the event) alongside the conditional probability of observing the evidence given that hypothesis.

In formulaic terms, Bayes' theorem is expressed as:

P(H|E) = [P(E|H) * P(H)] / P(E)

Here, P(H|E) represents the conditional probability of the hypothesis H given the evidence E. This relationship is crucial in fields such as healthcare statistics, where clinicians may need to revise their initial probabilities (such as the likelihood of a disease being present) as they gather more information from tests or patient histories.

While the addition rule pertains to the probabilities of the union of events and the multiplication rule relates to the joint probability of independent events, these do not fundamentally emphasize how conditional probabilities operate and are updated. The joint probability rule focuses on the probability of two events occurring simultaneously, but it lacks the dynamic aspect of incorporating new information as expressed in Bayes' theorem.