Conditional Probability and Independence
Conditional probability describes how the likelihood of one event changes once another event is known to have occurred, and independence describes the special case where knowing one event tells us nothing about another. These ideas, together with Bayes' theorem, explain how evidence updates beliefs and underlie the interpretation of diagnostic tests in medicine.
Definition
The conditional probability of event A given event B is the probability that A occurs when B is known to have occurred, defined as the probability of both A and B divided by the probability of B; A and B are independent if the conditional probability of A given B equals the unconditional probability of A.
Scope
The entry covers the definition of conditional probability, the multiplication rule, statistical independence, the law of total probability, and Bayes' theorem. It connects these to the appraisal of diagnostic tests, where the predictive value of a result depends on the prevalence of disease. It is a methodological reference, not clinical guidance on ordering or acting on specific tests.
Core questions
- How does knowing one event change the probability of another?
- When are two events independent, and what does that imply?
- How does Bayes' theorem reverse a conditional probability?
- Why does a positive test result mean different things at different prevalences?
Key concepts
- Conditional probability
- Multiplication rule
- Statistical independence
- Law of total probability
- Bayes' theorem
- Prior and posterior probability
- Prevalence and predictive value
- Sensitivity and specificity
Mechanisms
Conditioning on an event restricts attention to the outcomes consistent with it, so the conditional probability of A given B rescales the joint probability of A and B by the probability of B. Two events are independent when this conditioning leaves the probability unchanged, equivalent to their joint probability factoring into the product of the marginals. The law of total probability builds an event's probability from its conditional probabilities across a partition of the sample space, and Bayes' theorem inverts a conditional probability, expressing the probability of a cause given an observed effect in terms of the reverse conditional and the prior. In diagnostic testing this is why the probability that a patient with a positive result truly has the disease (the predictive value) depends not only on the test's sensitivity and specificity but also on the prior prevalence.
Clinical relevance
Conditional probability and Bayes' theorem describe how a test result revises the probability of disease, which is why identical tests yield different predictive values in high- and low-prevalence settings. This entry explains that reasoning as methodology and is not guidance for managing an individual patient.
History
The idea of updating probabilities in light of evidence is associated with Thomas Bayes, whose essay was communicated posthumously by Richard Price in 1763, and was generalised by Laplace. The resulting Bayes' theorem became central to statistics and, in the twentieth century, to the formal evaluation of diagnostic tests, where it links sensitivity, specificity, and prevalence to predictive value.
Key figures
- Thomas Bayes
- Richard Price
- Pierre-Simon Laplace
Related topics
Seminal works
- bayes-1763
- altman-bland-1994-diagnostic
- ross-2014
Frequently asked questions
- What is the difference between conditional probability and joint probability?
- Joint probability is the chance that two events both occur, while conditional probability is the chance that one occurs given that the other already has; the conditional probability equals the joint probability divided by the probability of the conditioning event.
- Why can a positive diagnostic test still mean the disease is unlikely?
- By Bayes' theorem the chance of disease after a positive result depends on prevalence; when a disease is rare, even an accurate test produces many false positives relative to true positives, so the predictive value of a positive result can be low.