Probability

--Probability experiments have outcomes.

--The set of all outcomes is the sample space, S.

--An event is any subset of the sample space. 
        Note:  It can be the whole sample space or the empty event.



Events and operations on events.

Let E and F be two events with sample space S.

--The compliment of event E, denoted as E , is all outcomes in S that are not in E.

--    "E and F" is a new event and         E and F = both event E and event F occur.
 
--    "E or F" is a new event and         E or F = event E occurs, event F occur, or both events occur.



Probability as a measurement

The probability of an event E, P(E), is a measurement of how often E occurs if one could repeat an experiment an infinite number of times.

In discrete probability experiments, P(E) = sum of the probabilities of all the outcomes in E.

An equally likely experiment is an experiment with a finite number of outcomes where each outcome has the same probability of occurring.



Probability rules

In equally likely experiments, P(E) = (number of events in E) / (number of events in S)

P(S) =1

P(empty set) =  0

Complimentation:  P(E) =  1 - P(E)

Addition Rule:  P(E or F) = P(E) + P(F) - P(E and F)

Two events are mutually exclusive if and only if P(E and F)  = 0.  Interpretation:  E and F can not both occur.

The conditional probability of E given F is defined as:  P(E | F) = P(E and F) / P(F).  It is assumed the P(F) isn't 0.

    This rule can be written as the multiplication rule:  P(E and F) = P(E | F) * P(F)