# Probability Theory/Independence

note to self: in the case of indep., the rules of the kind ${\displaystyle P(\sum \prod )=\sum \prod P}$ should be derived.


Definition (independence of events):

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space and let ${\displaystyle A,B\in {\mathcal {F}}}$. ${\displaystyle A}$ and ${\displaystyle B}$ are said to be independent iff ${\displaystyle P(A\cap B)=P(A)P(B)}$.

Remark (independence and conditional probability):

Using the definition of conditional probability, if e.g. ${\displaystyle P(A)\neq 0}$ we may rephrase the independence of ${\displaystyle A}$ and ${\displaystyle B}$ as ${\displaystyle P(B)=P(B|A)}$.