A random experiment can have several possible outcomes, but before the experiment is performed we do not know which one will occur. Probability gives a numerical way to describe that uncertainty.

The set of all possible outcomes is the sample space, usually written $\Omega$. An event is a subset of the sample space.

For example, when a six-sided die is rolled,

$$\Omega={1,2,3,4,5,6},$$

and the event of rolling an even number is

$$A={2,4,6}.$$

Probability values

A probability lies between $0$ and $1$:

$$0\le P(A)\le1.$$

An impossible event has probability $0$, while the whole sample space has probability $1$.

If two events cannot occur together, their probabilities add. More generally, set operations such as union, intersection and complement describe how events are combined.

Complements

The complement $A^c$ contains the outcomes for which $A$ does not occur. Because either $A$ or its complement must occur,

$$P(A^c)=1-P(A).$$

Equally likely finite outcomes

When a finite sample space consists of equally likely outcomes,

$$P(A)=\frac{|A|}{|\Omega|}.$$

For the die example,

$$P(A)=\frac36=\frac12.$$

This counting formula is useful, but it is not the definition of probability in general. Probability models can also assign unequal probabilities or describe continuous outcomes where individual outcomes cannot simply be counted.