.
Two events A,B are independent, if the conditional probability is the same as
unconditional, 
. This is stated in multiplicative form
which exhibits symmetry and includes trivial
events
Independence captures the intuition of non-interaction, and lack of information. In modeling it is often assumed rather than verified. For instance, we shall assume that the events generated by consecutive outputs of the random generator are independent. We also assume that tosses of a coin (fair, or not!) are independent.
Beginners sometimes confuse disjoint versus independent events. Exclusive (ie. disjoint) events capture the intuition of non-compatible outcomes. Not compatible outcomes cannot happen at the same time. This is not the same as independent outcomes. If A, B are disjoint and you know that A occurred, then you do know a lot about B. Namely you know that B cannot occur. Thus there is an interaction between A and B. Knowing whether A occurred influences chances of B, which is not possible under independence.
Independence (or, more properly, mutual stochastic independence) of
families of events is defined by requesting a much larger number of multiplicative
conditions. The reason behind is Theorem , which provides a very
convenient tool.
Another important concept is the conditional independence. For example, many events in the past and in the future are dependent. But in many mathematical models, past and future are independent conditionally on the present situation. In such a model future depends on past only through present events!
