Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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back to ProbabilityAxioms

:SeT -- CartesianProduct -- RandomVariable -- IndependentEvents

In any discussion of probabilities of events, we begin with a precise definition of all the possible events. This is what we call the '''sample space'''. In the case of tossing a coin whose sides are labeled ''heads'' and ''tails'', it would be improper to decide after experimentation has begun that the result ''neither heads nor tails'' has occured. That would be an event in a different sample space so we would be changing the rules of the game, so to speak.

For some kinds of experiments, there may be two or more '''independent''' sample spaces available. For example, when drawing a card from a standard bridge deck (of 52 cards from Ace to King in each of four suits), one sample space could be the denomination (Ace through King) and another could be the suit (clubs, diamonds, hearts, or spades). A complete description of an event would specify both the denomination and the suit. Such a sample space can be constructed as the CartesianProduct of the two independent sample spaces. 

Although the sample spaces are independent, some assignments of probabilities to points may lead to a circumstance in which the random variables thus defined are not describable as independent.
Similarly, it is possible (but rare) that two events defined on the same sample space might be classified as independent.