Probability Theory/Independence

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

Definition (independence of events):

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

Remark (independence and conditional probability):

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