 StatLect

Probability space

A probability space is a triple , where is a sample space, is a sigma-algebra of events and is a probability measure on . Elements of a probability space

The three building blocks of a probability space can be described as follows:

• the sample space is the set of all possible outcomes of a probabilistic experiment;

• the sigma-algebra is the set of all subsets of that are considered events;

• the probability measure is a function that associates a probability to each of the events belonging to the sigma-algebra .

Example

Suppose the probabilistic experiment consists in extracting a ball from an urn containing two balls, one red ( ) and one blue ( ).

The sample space is A possible sigma-algebra of events is where the event (the empty set) could be described as "nothing happens", the event could be described as "either a blue ball or a red ball is extracted", the event could be described as "a red ball is extracted", and the event could be described as "a blue ball is extracted".

A possible probability measure on is More details

The lecture entitled Probability contains a more detailed explanation of the concept of probability space and of the properties that must be satisfied by its building blocks.