Independent Events

A precise treatment of independence, conditional probability, pairwise versus mutual independence, and common misconceptions.

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Events are subsets of a sample space, and conditional probability changes the reference set. A sample-space rectangle contains two overlapping event regions and a small two-stage probability tree. The overlap marks joint occurrence, while branch labels show how later probabilities can depend on an earlier outcome.

Precise definition

Events $A$ and $B$ are independent when $P(A\cap B)=P(A)P(B)$. If $P(B)>0$, this is equivalent to $P(A\mid B)=P(A)$. Independence means learning one event occurred does not change the probability of the other under the model.

Notation and mathematical language

Disjoint events satisfy $P(A\cap B)=0$. Two disjoint events with positive probabilities are therefore not independent. For three events, pairwise independence does not guarantee mutual independence; mutual independence requires product rules for every subcollection of size at least two.

Conceptual picture

Independence is a property of a joint distribution, not of whether events look unrelated. Repeated trials may be modelled independent when the mechanism resets and no shared influence remains, but that assumption should be supported.

Fully worked example

Interpretation and application

Independence assumptions enable probability products, randomized trials, reliability calculations, and statistical models. Hidden common causes, clustering, and time dependence can invalidate them and make uncertainty estimates too small.