142. Independence

Two events are said to be independent if the occurrence of one event does not change our beliefs about the other

Events A and B are independent from each other if:

Implies:

Applies even if $𝑃\left(𝐴\right)=0$

142.1. Independence of Complements

If A and B are independent, then the following pairs are also independent:

• $𝐴$ and ${𝐵}^{𝑐}$
• ${𝐴}^{𝑐}$ and $𝐵$
• ${𝐴}^{𝑐}$ and ${𝐵}^{𝑐}$

Formally:

If

then:

Proof

Start with:

Since $\left(𝐴\cap 𝐵\right)$ and $\left(𝐴\cap {𝐵}^{𝑐}\right)$ are disjoint, by the Additivity Axiom:

Rearanging:

Thus we are back at out definition of independence:

142.2. Conditional Independence

142.3. Pairwise Independence

• Mutual (collective) independence $\Rightarrow$ pairwise independence

  If a collection of events is mutually independent, then every pair of events in the collection is independent.

• Pairwise independence $\Rightarrow$ mutual (collective) independence

  A collection of events can be pairwise independent but not mutually independent.

142.4. Reliability