AcademyProbability
Academy
Events
Level 1 - Math II (Physics) topic page in Probability.
Principle
Events are subsets of a sample space. Set operations describe how events combine: not happening, one event or another happening, both happening, or one happening without the other.
Notation
\(S\)
sample space: all outcomes
\(A,B\)
events, so A and B are subsets of S
\(A^c\)
complement: outcomes in S that are not in A
\(A\cup B\)
union: outcomes in A or B or both
\(A\cap B\)
intersection: outcomes in both A and B
\(A\setminus B\)
difference: outcomes in A but not in B
\(\varnothing\)
empty set: event with no outcomes
\(A\subseteq B\)
subset: every outcome in A is also in B
Method
Step 1: Name the sample space
Write \(S\) before translating statements. The complement \(A^c\) always means complement inside this chosen \(S\).
Step 2: Translate keywords
Use union for inclusive or, intersection for and, complement for not, and difference for in one event but not another.
Step 3: Simplify the set statement
Check whether events overlap, whether one event is contained in another, and whether the expression can be rewritten using set laws.
Not A
\[A^c=S\setminus A\]
A or B
\[A\cup B\]
A and B
\[A\cap B\]
A but not B
\[A\setminus B=A\cap B^c\]
Rules
Complement
\[A^c=S\setminus A\]
Union
\[A\cup B=\{s\in S:s\in A\text{ or }s\in B\}\]
Intersection
\[A\cap B=\{s\in S:s\in A\text{ and }s\in B\}\]
Difference
\[A\setminus B=A\cap B^c\]
Disjoint events
\[A\cap B=\varnothing\]
Subset
\[A\subseteq B\quad\Longleftrightarrow\quad s\in A\Rightarrow s\in B\]
De Morgan 1
\[(A\cup B)^c=A^c\cap B^c\]
De Morgan 2
\[(A\cap B)^c=A^c\cup B^c\]
Examples
Question
For one die roll, let \(A\) be even and \(B\) be at least 5. Write \(A\), \(B\),
\[A\cup B\]
and \[A\cap B\]
Answer
With
\[S=\{1,2,3,4,5,6\}\]
\[A=\{2,4,6\}\]
and \[B=\{5,6\}\]
Then \[A\cup B=\{2,4,5,6\}\]
and \[A\cap B=\{6\}\]
Checks
- In probability, or normally means inclusive or: \(A\cup B\) includes outcomes where both events occur.
- And means intersection: \(A\cap B\).
- Disjoint events cannot occur together because their intersection is empty.
- A complement depends on the sample space \(S\).