Independence (probability theory)

Two events are independent when their joint probability equals the product of their probabilities. Equivalently, learning that one occurred does not change the probability of the other, when conditioning is defined.

Conditional probability: The probability of an event given that another event has occurred. Independence means conditioning on either event leaves the other event’s probability unchanged.
Pairwise independence: A property in which every pair of events in a collection is independent, without requiring larger groups to be independent. It is weaker than mutual independence, despite guaranteeing independence for each pair.
Binomial distribution: The distribution of the number of successes in a fixed number of independent Bernoulli trials. Its probability formula relies on repeated trials with the same success probability and independence.
Independent coin flips: A sequence of coin-toss outcomes modeled so each toss has a fixed probability unaffected by earlier tosses. This standard model makes independence concrete through repeated trials.
Joint probability: The probability that two or more events occur together. For independent events, their joint probability is the product of their individual probabilities.
Conditional independence: A relation in which events are independent after conditioning on a specified event or variable. Events can be dependent overall yet independent once additional information is known.
Law of large numbers: A theorem stating that sample averages converge toward an expected value under specified conditions. Independence is a common condition ensuring averages of repeated observations stabilize.
Three pairwise independent events: A classic construction of three events that are independent in pairs but not mutually independent. It demonstrates directly why pairwise independence does not imply mutual independence.
Mutual independence: A condition requiring every finite combination of events in a collection to satisfy the independence product rule. It extends pairwise independence to whole collections, including combinations of three or more events.
Correlation: A standardized measure of the direction and strength of linear association between two variables. Zero correlation can hold without independence, except under special distributional conditions.

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