The sample space $\Omega$ is the set of every possible result of a random experiment. Each single result $\omega \in \Omega$ is called an outcome.
Rolling a die gives $\Omega = \{1,2,3,4,5,6\}$, where each number is one outcome.
An event $A$ is any subset of the sample space, $A \subseteq \Omega$. It occurs when the outcome of the experiment belongs to $A$.
| Symbol | Meaning |
|---|---|
| $\Omega$ | sample space, the set of all outcomes |
| $\omega$ | a single outcome |
| $A, B, \dots$ | events, subsets of $\Omega$ |
| $\emptyset$ | impossible event, the empty subset |
| $P(A)$ | probability of event $A$ |
| $A^c$ | complement of $A$, meaning $\Omega$ without $A$ |
| $A \cap B$ | intersection, outcomes in both $A$ and $B$ |
| $A \cup B$ | union, outcomes in $A$ or $B$ or both |
Six equally likely outcomes, one of them is a $1$.
Two equally likely outcomes, one of them is heads.
These three axioms are enough to derive every other probability rule, including the ones below.
The complement $A^c$ is the event that $A$ does not occur. Since $A$ and $A^c$ together always cover $\Omega$ and never overlap, their probabilities add up to $1$.
$A \cap B$ is the event that both $A$ and $B$ occur. $A \cup B$ is the event that at least one of $A$ or $B$ occurs.
Adding $P(A)$ and $P(B)$ counts the overlap twice, so it has to be subtracted once.
Set $P(A)$, $P(B)$ and $P(A \cap B)$ with the sliders. The diagram and $P(A \cup B)$ update to match, drawn to scale by area.