Independence of events

Independent pairs

We take our clue from the condition $P\left(A\right|B)=P(A)$ . Property (CP4) for conditional probability (in the case of equality) yields sixteen equivalent conditions as follows.

These conditions are equivalent in the sense that if any one holds, then all hold. We may chose any one of these as the defining condition and consider the others as equivalentsfor the defining condition. Because of its simplicity and symmetry with respect to the two events, we adopt the product rule in the upper right hand corner of the table.

Definition. The pair $\{A,B\}$ of events is said to be (stochastically) independent iff the following product rule holds:

Remark . Although the product rule is adopted as the basis for definition, in many applications the assumptions leading to independence may be formulated more naturally interms of one or another of the equivalent expressions. We are free to do this, for the effect of assuming any one condition is to assume them all.

The equivalences in the right-hand column of the upper portion of the table may be expressed as a replacement rule , which we augment and extend below:

If the pair $\{A,B\}$ independent, so is any pair obtained by taking the complement of either or both of the events.

We note two relevant facts

• Suppose event N has probability zero (is a null event). Then for any event A , we have $0\le P\left(AN\right)\le P\left(N\right)=0=P\left(A\right)P\left(N\right)$ , so that the product rule holds. Thus $\{N,A\}$ is an independent pair for any event A .
• If event S has probability one (is an almost sure event), then its complement S ^c is a null event. By the replacement rule and the fact just established, $\{S^c,A\}$ is independent, so $\{S,A\}$ is independent.

The replacement rule may thus be extended to:

Replacement rule

Caution

1. Unless at least one of the events has probability one or zero, a pair cannot be both independent and mutually exclusive. Intuitively, if the pair is mutually exclusive, thenthe occurrence of one requires that the other does not occur. Formally: Suppose $0<P\left(A\right)<1$ and $0<P\left(B\right)<1$ . $\{A,B\}$ mutually exclusive implies $P\left(AB\right)=P\left(\varnothing \right)=0\ne P\left(A\right)P\left(B\right)$ . $\{A,B\}$ independent implies $P\left(AB\right)=P\left(A\right)P\left(B\right)>0=P\left(\varnothing \right)$
2. Independence is not a property of events. Two non mutually exclusive events may be independent under one probability measure, but may not be independent for another.This can be seen by considering various probability distributions on a Venn diagram or minterm map.

Independent classes

Extension of the concept of independence to an arbitrary class of events utilizes the product rule.

Definition. A class of events is said to be (stochastically) independent iff the product rule holds for every finite subclass of two or more events in the class.

A class $\{A,B,C\}$ is independent iff all four of the following product rules hold