Inclusion-Exclusion Principle/Examples/3 Events in Event Space: Example
Jump to navigation
Jump to search
Examples of Use of Inclusion-Exclusion Principle
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 $\dfrac 6 {10}$.
Proof
We are looking for the probability of:
- $\paren {\paren {A \cap B} \setminus \paren {A \cap B \cap C} } \cup \paren {\paren {B \cap C} \setminus \paren {A \cap B \cap C} } \cup \paren {\paren {A \cap C} \setminus\paren {A \cap B \cap C} }$
We have that:
- $\paren {\paren {A \cap B} \setminus \paren {A \cap B \cap C} } \cap \paren {A \cap B \cap C} = \O$
and similarly for the other two such terms.
Then we have that $\paren {A \cap B} \setminus \paren {A \cap B \cap C}$, $\paren {\paren {B \cap C} \setminus \paren {A \cap B \cap C} }$ and $\paren {\paren {A \cap C} \setminus \paren {A \cap B \cap C} }$ are pairwise disjoint.
Hence the probability $P$ that exactly $2$ of the events $A$, $B$ and $C$ occur is
\(\ds P\) | \(=\) | \(\ds \paren {\map \Pr {A \cap B} - \map \Pr {A \cap B \cap C} } - \paren {\map \Pr {B \cap C} - \map \Pr {A \cap B \cap C} } - \paren {\map \Pr {A \cap C} - \map \Pr {A \cap B \cap C} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac 3 {10} - \dfrac 1 {10} } - \paren {\dfrac 4 {10} - \dfrac 1 {10} } - \paren {\dfrac 2 {10} - \dfrac 1 {10} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 6 {10}\) |
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.4$: Probability spaces: Exercise $9$