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 $\dfrac 6 {10}$.