Inclusion-Exclusion Principle/Examples
Jump to navigation
Jump to search
Examples of Use of Inclusion-Exclusion Principle
$3$ Events in Event Space
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $A, B, C \in \Sigma$.
Then:
\(\ds \map \Pr {A \cup B \cup C}\) | \(=\) | \(\ds \map \Pr A + \map \Pr B + \map \Pr C\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \map \Pr {A \cap B} - \map \Pr {B \cap C} - \map \Pr {A \cap C}\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \map \Pr {A \cap B \cap C}\) |
$3$ Events in Event Space: Example
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $A, B, C \in \Sigma$ such that:.
Then:
\(\ds \map \Pr A\) | \(=\) | \(\ds \dfrac 5 {10}\) | ||||||||||||
\(\ds \map \Pr B\) | \(=\) | \(\ds \dfrac 7 {10}\) | ||||||||||||
\(\ds \map \Pr C\) | \(=\) | \(\ds \dfrac 6 {10}\) | ||||||||||||
\(\ds \map \Pr {A \cap B}\) | \(=\) | \(\ds \dfrac 3 {10}\) | ||||||||||||
\(\ds \map \Pr {B \cap C}\) | \(=\) | \(\ds \dfrac 4 {10}\) | ||||||||||||
\(\ds \map \Pr {A \cap C}\) | \(=\) | \(\ds \dfrac 2 {10}\) | ||||||||||||
\(\ds \map \Pr {A \cap B \cap C}\) | \(=\) | \(\ds \dfrac 1 {10}\) |
The probability that exactly $2$ of the events $A$, $B$ and $C$ occur is $\dfrac 6 {10}$.