Event

We say an event \(A\) occurs when the outcome of the experiment belongs to \(A\). On a die, «an even number shows» is \(A=\{2,4,6\}\). An event with a single outcome is called elementary; \(\Omega\) is the sure event and \(\varnothing\) the impossible one.

Since events are sets, they combine through set operations: the union \(A\cup B\) is «\(A\) or \(B\) occurs», the intersection \(A\cap B\) is «both occur» and the complement \(A^c\) is «\(A\) does not occur». With \(A=\{2,4,6\}\) and \(B=\{4,5,6\}\) we get \(A\cap B=\{4,6\}\) and \(A\cup B=\{2,4,5,6\}\). If \(A\cap B=\varnothing\), the events are mutually exclusive.

If \(\Omega\) has \(n\) elements, there are \(2^n\) possible events, because each outcome is either in the subset or not. That is why, in large spaces, it is better to describe events by a condition than to list their elements.

A frequent mistake is to call a single outcome an «event» or to confuse «mutually exclusive» with «independent»; these are different ideas, and we will meet them later. Assigning a number to each event is the topic of the next concept.