Example:Sample space for the sum of two dice

From CS2800 wiki

Consider an experiment where two fair dice are rolled independently, and their values are added. We might model this experiment with the sample space [math]\href{/cs2800/wiki/index.php/S}{S} = \href{/cs2800/wiki/index.php/Enumerated_set}{\{2,3,\dots,12\}} [/math] (or even [math]\href{/cs2800/wiki/index.php/S}{S} = \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math]), or with the sample space [math]\href{/cs2800/wiki/index.php/S}{S} = \href{/cs2800/wiki/index.php/Enumerated_set}{\{1,\dots,6\}} \href{/cs2800/wiki/index.php?title=%E2%A8%AF&action=edit&redlink=1}{⨯} \href{/cs2800/wiki/index.php/Enumerated_set}{\{1,\dots,6\}} = \href{/cs2800/wiki/index.php/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 outcomes 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 events of interest: for example the sample space [math]\href{/cs2800/wiki/index.php/S}{S} = \{x\} [/math] wouldn't work for this experiment, since there is no way to interpret the sum of the dice.