Inclusion-Exclusion Principle/Examples/3 Events in Event Space: Example

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:

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