# Inclusion-Exclusion Principle/Examples

## 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 $\dfrac 6 {10}$.