Chance, outcomes and complements

An outcome is a possible result of an experiment; an event is a collection of outcomes. Probability lies between \(0\) and \(1\). In a finite equally likely sample space, \(\mathrm P(A)=\dfrac{\text{number of outcomes in}\hspace{0.5em}A}{\text{total number of outcomes}}\). State why the outcomes are equally likely before using this ratio.

Example

A fair six-sided die has outcomes \(\{1,2,3,4,5,6\}\). Prime scores form \(A=\{2,3,5\}\), so \(\mathrm P(A)=\dfrac36=\dfrac12\). The number \(1\) is not prime.

The complement \(A^{\prime}\) means “not \(A\)”. Since the event and its complement cover all outcomes without overlap, \(\mathrm P(A^{\prime})=1-\mathrm P(A)\). Thus the probability of not a prime is also \(\dfrac12\).

For this finite model, an impossible event has probability \(0\) and a certain event has probability \(1\). A biased device need not give equally likely outcomes; simply listing six faces does not justify assigning each probability \(\dfrac16\).