Suppose that \(\displaystyle \Omega\) is a set of points and \(\displaystyle \mathscr{Pr}\) is a function on the powerset \(\displaystyle \mathcal{P}(\Omega).\) Here are Morris Marx's axioms for probability.

**Axiom 1.** If \(\displaystyle A\in\mathcal{P}(\Omega).\) Then \(\displaystyle A\) is an event and \(\displaystyle \mathscr{Pr}(A)\ge 0\)

**Axiom 2. **\(\displaystyle \mathscr{Pr}(\Omega)=1\)

**Axiom 3. **If \(\displaystyle \{A,B\}\subset\mathcal{P}(\Omega)~\&~A\cap B=\emptyset\) then \(\displaystyle \mathscr{Pr}(A\cup B)=\mathscr{Pr}(A)+\mathscr{Pr}( B)\)

Note that \(\displaystyle A\cap A^c=\emptyset~\&~A\cup A^c=\Omega\)

\(\displaystyle \mathscr{Pr}(A\cup A^c)=\mathscr{Pr}(A)+\mathscr{Pr}( A^c)=\mathscr{Pr}(\Omega)=1\)

Thus \(\displaystyle \mathscr{Pr}(A^c)=1-\mathscr{Pr}(A)\)

Because \(\displaystyle \Omega^c=\emptyset\) show that \(\displaystyle \mathscr{Pr}(\emptyset)=0.\)

If \(\displaystyle A\subset B\) then show that \(\displaystyle \mathscr{Pr}(A)\le\mathscr{Pr}(B)\)

For any event \(\displaystyle A\) then show that \(\displaystyle \mathscr{Pr}(A)\le 1\)

For any events \(\displaystyle A~\&~B\) show that \(\displaystyle \mathscr{Pr}(A\cup B)=\mathscr{Pr}(A)+\mathscr{Pr}(B)-\mathscr{Pr}(A\cap B)\)