I am going to calculate this probability \begin{align} P=\Pr\left\{ \frac{1+X_1}{1+Y}> a \cap \frac{1+\frac{X_2}{X_2+1}}{1+\frac{Y}{Y+1}} >b \right\} \end{align} I tried to calculate as follows: \begin{align} P&=\int \limits_{0}^{\infty} \Pr\left\{ \frac{1+X_1}{1+y}> a \cap \frac{1+\frac{X_2}{X_2+1}}{1+\frac{y}{y+1}} >b \right\} f_{Y} (y) dy \\ &=\int \limits_{0}^{\infty} \Pr\left\{ \frac{1+X_1}{1+y}> a \right\} \Pr \left\{ \frac{1+\frac{X_2}{X_2+1}}{1+\frac{y}{y+1}} >b \right\} f_{Y} (y) dy \end{align} Here, $X_i$, $Y$ are exponential random variable with mean value $\Omega_{X_i}$ and $\Omega_{Y}$, respectively. However, when I do Monte Carlo simulation, it does not match well. Do you know what is the reason and is there any way to find this probability?
Thank you!