Given that $X,Y,Z$ are exponential random variable with the corresponding parameters ${\lambda _1},{\lambda _2},{\lambda _3}$, does the following derivation of this probability $A=\Pr[ {X > \frac{2}{Y} \cap X > \frac{4}{Z}}]$ correct ?
$\begin{gathered} \,\,\,\,\Pr \left[ {X > \frac{2}{Y} \cap X > \frac{4}{Z}} \right] \hfill \\ = \Pr \left[ {X > \max \left[ {\frac{2}{Y},\frac{4}{Z}} \right]} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \Pr \left[ {X > \frac{2}{Y} \cap \frac{2}{Y} > \frac{4}{Z}} \right] + \Pr \left[ {X > \frac{4}{Z} \cap \frac{4}{Z} > \frac{2}{Y}} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \Pr \left[ {\frac{4}{Z} < \frac{2}{Y} < X} \right] + \Pr \left[ {\frac{2}{Y} < \frac{4}{Z} < X} \right] \hfill \\ \end{gathered} $
Where $\Pr \left[ {\frac{4}{Z} < \frac{2}{Y} < X} \right] = \Pr \left[ {\frac{2}{Y} < X} \right] - \Pr \left[ {\frac{2}{Y} < \frac{4}{Z}} \right]$ (First point: Am I wrong here ?)
and $\Pr \left[ {\frac{2}{Y} < \frac{4}{Z} < X} \right] = \Pr \left[ {\frac{4}{Z} < X} \right] - \Pr \left[ {\frac{4}{Z} < \frac{2}{Y}} \right]$ (Second point: Am I wrong here ?)
So $A = \Pr \left[ {\frac{2}{Y} < X} \right] + \Pr \left[ {\frac{4}{Z} < X} \right] - \left( {\Pr \left[ {\frac{2}{Y} < \frac{4}{Z}} \right] + \Pr \left[ {\frac{4}{Z} < \frac{2}{Y}} \right]} \right) = \Pr \left[ {\frac{2}{Y} < X} \right] + \Pr \left[ {\frac{4}{Z} < X} \right] - 1$