Independent Events
==================


.. proof:definition:: Independent Events

   We define two events $A$ and $B$ to be **independent** in case:

   .. math::
      \P(A\,B) = \P(A)\,\P(B)

   and we often write $A\perp B$ to denote independent events.


When two events are independent it is simple to show that the
conditional probability $\P(A\given B)$ equals $\P(A)$ i.e. knowledge
about $B$ does not influence the probability of $A$.  Evidently we
also have $A\perp B\Longrightarrow\P(B\given A)=\P(B)$ .

A well known example is throwing with two dice. The outcome of the
first die in no way influences the outcome of throwing with the
second die. Therefore throwing 2 times 6 in a row is $1/6\times 1/6 =
1/36$ .

An example of dependent events can be found in the previous section
about the marbles from the vases. The events ´Vase 1' and 'Red' are
evidently not independent.


Now consider events $A$ and $B$ and a third event $C$.

.. proof:definition:: Conditional Independence

   Events $A$ and $B$ given $C$ are conditional independent in case:

   .. math::
      \P(AB\given C) = \P(A\given C) \P(B\given C)

Independence of $A$ and $B$ does not imply conditional independence or
vice versa.