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In probability theory, a probability space or a probability triple {\displaystyle } is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die.
A probability space consists of three elements:
In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article.
In the example of the throw of a standard die, we would take the sample space to be { 1 , 2 , 3 , 4 , 5 , 6 } {\displaystyle \{1,2,3,4,5,6\}}. For the event space, we could simply use the set of all subsets of the sample space, which would then contain simple events such as { 5 } {\displaystyle \{5\}} , as well as complex events such as { 2 , 4 , 6 } {\displaystyle \{2,4,6\}} . Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 — so for example, { 5 } {\displaystyle \{5\}} would be mapped to 1 / 6 {\displaystyle 1/6} , and { 2 , 4 , 6 } {\displaystyle \{2,4,6\}} would be mapped to 3 / 6 = 1 / 2 {\displaystyle 3/6=1/2}.