View source for Example:Sample space for the sum of two dice

Consider an experiment where two [[fair]] [[dice]] are rolled [[independent]]ly, and their values are added.  We might model this experiment with the sample space <math>[[S]] = [[Enumerated set|\{2,3,\dots,12\}]]</math> (or even <math>[[S]] = [[ℝ]]</math>), or with the sample space <math>[[S]] = [[Enumerated set|\{1,\dots,6\}]] [[⨯]] [[Enumerated set|\{1,\dots,6\}]] = [[Enumerated set|\{(1,1),(1,2),\dots,(2,1),\dots,(6,6)\}]]</math>.

Either [[sample space]] would work, since both contain enough information to describe the [[outcome]]s of the experiment.  However, it is difficult to describe the [[probability measure]] with the first model, while it is easy with the second (for the second, the fact that the dice are [[fair]] and [[independent]] means that the [[equiprobable measure]] is a good [[probability measure]] to describe the experiment).

There is not a "correct" [[sample space]] for a given problem, but there are some that are easier to work with than others.  It is also possible to create a [[sample space]] that doesn't have enough resolution to interpret the [[event]]s of interest: for example the sample space <math>[[S]] = \{x\}</math> wouldn't work for this experiment, since there is no way to interpret the sum of the [[dice]].