# Two independent events

$$\newcommand{\bP}{\mathbb{P}}$$

We say that two events, $$A$$ and $$B$$, are independent when learning that $$A$$ has occurred does not change your probability that $$B$$ occurs. That is, $$\bP(B \mid A) = \bP(B)$$. Equivalently, $$A$$ and $$B$$ are independent if $$\bP(A)$$ doesn’t change if you condition on $$B$$: $$\bP(A \mid B) = \bP(A)$$.

Another way to state independence is that $$\bP(A,B) = \bP(A) \bP(B)$$.

All these definitions are equivalent:

$$\bP(A,B) = \bP(A)\; \bP(B \mid A)$$

by the chain rule, so

$$\bP(A,B) = \bP(A)\; \bP(B)\;\; \Leftrightarrow \;\; \bP(A)\; \bP(B \mid A) = \bP(A)\; \bP(B) \ ,$$

and similarly for $$\bP(B)\; \bP(A \mid B)$$.

Children:

Parents:

• Probability theory

The logic of science; coherence relations on quantitative degrees of belief.

• I’m not sure what this equation is trying to tell me. Some parts of it are only true if A and B are independent, but some parts are true in general, right?

• Yeah this is maybe a placeholder to support the lens, added stub tag.