Independent Events

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.

Conditions and key results

Conditional probabilities require positive conditioning probability. Sampling without replacement is generally dependent; with replacement can be independent under identical random draws. Empirical near-equality does not prove exact population independence.

A reliable strategy

1. Define events and compute marginal and intersection probabilities from a joint table, tree, or model.
2. Compare $P(A\cap B)$ with $P(A)P(B)$ or compare a conditional probability with its marginal.
3. For more than two events, test the required intersections rather than only pairs.
4. State whether independence is assumed by design, established mathematically, or only assessed from sample evidence.

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.

Common mistakes

Verification and reasonableness

• Compute the same joint probability from a conditional tree and a product formula.
• Check complements: if $A,B$ are independent, so are $A^c,B$ and other complement pairs.
• Simulate or inspect stratified data for dependence, while distinguishing evidence from proof.

Practice

1. If $P(A)=0.4,P(B)=0.5$ and events are independent, find $P(A\cap B)$.
2. Can positive-probability disjoint events be independent?
3. Does pairwise independence imply mutual independence?

Further deduction

A standard pairwise-not-mutual example uses two independent fair coin bits $X,Y$ and $Z=X\oplus Y$. Each pair is independent, but knowing any two determines the third, and $P(X=0,Y=0,Z=0)=1/4\ne1/8$. This demonstrates why higher-order intersection conditions carry information absent from pairwise checks.

If $A$ and $B$ are independent, their indicators have covariance zero: $E[I_AI_B]-E[I_A]E[I_B]=P(A\cap B)-P(A)P(B)=0$. Zero covariance for arbitrary numeric variables does not imply independence, except under special families such as jointly normal variables. Event independence and uncorrelatedness should therefore not be generalized carelessly.

Related topics

• Conditional Probability
• Bayes' Theorem
• Probability