# Example:Sample space for a six-sided die

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 . The assumption that the die is fair means that . Using the second and third Kolmogorov axioms, we see that these probability are all .

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 is a finite sample space, and is a function satisfying then there is a unique probability measure having for all .

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.