PROBABILITY 02
Outcomes and Events
An outcome is one possible result; an event is a set of outcomes whose occurrence we want to study.
Outcomes
If a die is rolled, an individual outcome may be \(4\). All possible outcomes form
\[\Omega=\{1,2,3,4,5,6\}.\]1. Events
An event is a subset of the sample space. The event “an even number occurs” is
\[A=\{2,4,6\}.\]The event occurs whenever the realised outcome belongs to \(A\).
2. Certain and impossible events
The entire sample space \(\Omega\) is the certain event, while the empty set \(\varnothing\) is the impossible event.
\[P(\Omega)=1,\qquad P(\varnothing)=0.\]3. Union
\(A\cup B\) means that \(A\), \(B\), or both occur.
4. Intersection
\(A\cap B\) means that both events occur.
5. Complement
The complement \(A^c\) contains outcomes for which \(A\) does not occur.
\[P(A^c)=1-P(A).\]6. Mutually exclusive events
If two events cannot occur together,
\[A\cap B=\varnothing.\]They are called mutually exclusive or disjoint.
7. Why set notation matters
Probability calculations are built on relationships between sets of outcomes.