Show that $\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}$

Question

Let $A$ and $B$ be independent events. Show that \begin{align*} \max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9} \end{align*}

MY ATTEMPT

Since $\textbf{P}(A\cap B) = \textbf{P}(A)\textbf{P}(B)$, we have

\begin{align*} \textbf{P}((A\cup B)^{c}) & = 1 - \textbf{P}(A\cup B) = 1 - \textbf{P}(A) - \textbf{P}(B) + \textbf{P}(A\cap B)\\ & = 1 - \textbf{P}(A) - \textbf{P}(B) + \textbf{P}(A)\textbf{P}(B)\\ & = (1 - \textbf{P}(A)) - \textbf{P}(B)(1 - \textbf{P}(A))\\ & = (1 - \textbf{P}(A))(1-\textbf{P}(B)) = \textbf{P}(A^{c})\textbf{P}(B^{c}) \end{align*}

Analagously, we get \begin{align*} \textbf{P}(A\triangle B) & = \textbf{P}(A) + \textbf{P}(B) - 2\textbf{P}(A\cap B) = \textbf{P}(A) + \textbf{P}(B) - 2\textbf{P}(A)\textbf{P}(B)\\\\ & = \textbf{P}(A)(1 - \textbf{P}(B)) + \textbf{P}(B)(1-\textbf{P}(A)) = \textbf{P}(A)\textbf{P}(B^{c}) + \textbf{P}(A^{c})\textbf{P}(B) \end{align*}

However, I do not know how to proceed from here. Am I on the right track? Could someone complete the proof? Any help is appreciated. Thanks in advance.

Answer 1

As in the comment by Lord Shark The Unknown you can first set $$ P(A) = p$$ and $$P(B) = q.$$ Then the thesis becomes the following.

Show that the inequalities \begin{eqnarray} pq &<& \frac{4}{9}\\ (1-p)(1-q) &<& \frac{4}{9}\\ p(1-q) + q(1-p) &<& \frac{4}{9} \end{eqnarray} cannot hold simultaneously.

Performing the following change of variables \begin{equation} \begin{cases} p = \frac{1}{2}(Q+P+1)\\ q = \frac{1}{2}(Q-P+1) \end{cases} \end{equation} leads to the corresponding inequalities in terms of $P$ and $Q$, i.e. \begin{eqnarray} (Q+1)^2 -P^2 &<& \frac{16}{9}\tag{1}\label{uno}\\ (Q-1)^2 -P^2 &<& \frac{16}{9}\tag{2}\label{due}\\ Q^2-P^2>\frac{1}{9}\tag{3}\label{tre} \end{eqnarray} which are areas delimited by hyperbolae (in red, green, and black respectively in the Figure). The red dashed area corresponds to the set of points $(P,Q)$ satisfying conditions \eqref{uno} and \eqref{due}. The black dashed areas mark te set of points that satisfy condition \eqref{tre}. Therefore the three inequalities cannot be all true. And the thesis follows.