Example:Sample space for a six-sided die

From CS2800 wiki

Suppose we wished to model an experiment where a single fair die is rolled (unless specificied otherwise, I will assume that all dice are six-sided).

We could model this experiment with a sample space [math]\href{/cs2800/wiki/index.php/S}{S} = \href{/cs2800/wiki/index.php/Enumerated_set}{\{1,2,3,4,5,6\}} [/math]. The assumption that the die is fair means that [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(\{1\}) = \href{/cs2800/wiki/index.php/Pr}{Pr}(\{2\}) = \cdots = \href{/cs2800/wiki/index.php/Pr}{Pr}(\{6\}) [/math]. Using the second and third Kolmogorov axioms, we see that these probability are all [math]1/6 [/math].

Note that for a finite sample space, it suffices to give the probabilities of the simple events: the events containing only a single outcome. This is justified by the following claim:

If [math]\href{/cs2800/wiki/index.php/S}{S} [/math] is a finite sample space, and [math]P : \href{/cs2800/wiki/index.php/S}{S} → \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math] is a function satisfying
  1. [math]P(s) ≥ 0 [/math] for all [math]s \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/S}{S} [/math]
  2. [math]\sum_{s \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/S}{S}} P(s) = 1 [/math]
then there is a unique probability measure [math]\href{/cs2800/wiki/index.php/Pr}{Pr} [/math] having [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(\{s\}) = P(s) [/math] for all [math]s [/math].

The proof is left as an exercise.

Note that by choosing this sample space, we are already ruling out the possibility that the die could land on a corner or roll off the table; it is important to be aware that the choice of model can affect the conclusions drawn using it.