Problem Setup
Let $R_a$ and $R_g$ be two independent random variable. $\forall i\in\{a,g\}$, the pdf of $R_i$ is given by $$ f_{R_i}(r_i)=2 \pi \lambda_i r_i e^{-\pi \lambda_i r_i^2} \quad \text{with} \,\, r_i \geq 0, \tag {1} $$ where $\lambda_i$ is positive real numbers.
Define $P_{r,g}=P_g(R_{g}^2+h_g^2)^{-\eta_g/2}$ and \begin{equation} P_{r,a}= \begin{cases} P_{r,{a, L}}= P_a(R_{a}^2+h_a^2)^{-\eta_{a, L}/2} & \text{with probability } P_L(R_a)\\ P_{r,{a, N}}= P_a(R_{a}^2+h_a^2)^{-\eta_{a, N}/2} & \text{with probability } P_N(R_a) \end{cases} \end{equation} where $P_i,h_i,\eta_i$ are positive real numbers, $P_N(r_a)=1-P_L(r_a)$ with \begin{equation} P_L(r_a)=\frac{1}{1 + a \exp \left( b \left[ a - \theta(r_a) \right] \right) } \end{equation} where $\theta(r_a)=\frac{180}{\pi} \arctan \left({h_a}/{r_a}\right)$, and $a, b$ are positive constants.
Question
Find $f_{X_a}(x_a)$ where $X_a$ is $R_a$ given that it satisfies $P_{r,a} >P_{r,g}$ with $P_{r,a}=P_{r,a, L}$.
Step #1: express the pdf of $R_a$ \begin{equation} f_{X_a}(x_a)= \frac{{\rm d}}{{\rm d}x_a} F_{X_a}(x_a) = - \frac{{\rm d}}{{\rm d}x_a} \mathbb{P}[{X_a} > x_a] \tag{2} \end{equation} with \begin{equation} \mathbb{P}[X_a>x_a]=\mathbb{P}[R_{a}>x_a\mid P_{r,a} >P_{r,g}]=\frac{\mathbb{P}[R_a>x_a,P_{r,a} >P_{r,g}]}{\mathbb{P}[P_{r,a} >P_{r,g}]}, \end{equation}
Step #2: express $\mathbb{P}[P_{r,a} >P_{r,g}]$
Let $p_a=\mathbb{P}[P_{r,a} >P_{r,g}]$, then $$ p_a=\mathbb{P}[P_{r,a} >P_{r,g} \mid P_{r,a}=P_{r,a,L}]=\frac{\mathbb{P}[P_{r,a} >P_{r,g}, P_{r,a}=P_{r,a,L}]}{\mathbb{P}[P_{r,a}=P_{r,a,L}]} \tag{3} $$
Step #3: compute $\mathbb{P}[P_{r,a}=P_{r,a,L}]$ \begin{eqnarray} \mathbb{P}[P_{r,a}=P_{r,a,L}] &= & \mathbb{P} \left[ P_L(R_a) \geq P_N(R_a) \right] \nonumber \\ &= & \mathbb{P} \left[ P_L(R_a) \geq 1- P_L(R_a) \right] \nonumber \\ &= & \mathbb{P} \left[ P_L(R_a) \geq \frac{1}{2} \right] \nonumber \\ % &= &\mathbb{P} \left[\frac{1}{1 + a \exp \left( b \left[ a - \theta(r_a) \right] \right)} \geq \frac{1}{2} \right] \nonumber \\ %&= & \mathbb{P} \left[ 1 + a \exp \left( b \left[ a - \theta(r_a) \right] \right) \leq 2 \right] \nonumber \\ %&= & \mathbb{P} \left[ a \exp \left( b \left[ a - \theta(r_a) \right] \right) \leq 1 \right] \nonumber \\ %&= & \mathbb{P} \left[ \exp \left( b \left[ a - \theta(r_a) \right] \right) \leq \frac{1}{a} \right] \nonumber \\ %&= & { \mathbb{P} \left[ b \left[ a - \theta(r_a) \right] \leq \ln\left(\frac{1}{a}\right) \right] }\nonumber \\ %&= & { \mathbb{P} \left[ b \left[ a - \theta(r_a) \right] \leq \ln(1) - \ln(a) \right] }\nonumber \\ %&= & { \mathbb{P} \left[ b \left[ a - \theta(r_a) \right] \leq - \ln(a) \right] }\nonumber \\ %&= & { \mathbb{P} \left[ a - \theta(r_a) \leq - \frac{\ln(a)}{b} \right] }\nonumber \\ %&= & { \mathbb{P} \left[ \theta(r_a) \geq a+ \frac{\ln(a)}{b} \right] }\nonumber \\ &= & { \mathbb{P} \left[ R_a \leq \frac{h_a}{\tan\left(\frac{\pi}{180}\left[a+ \frac{\ln(a)}{b}\right]\right)} \right] }\nonumber \\ &=& F_{R_a}\left( \frac{h_a}{\tan\left(\frac{\pi}{180}\left[a+ \frac{\ln(a)}{b}\right]\right)} \right) \tag{4} \end{eqnarray}
Step #4: compute $\mathbb{P}[P_{r,a} >P_{r,g}, P_{r,a}=P_{r,a,L}]$ \begin{eqnarray} &&\mathbb{P}[P_{r,a} >P_{r,g}, P_{r,a}=P_{r,a,L}] \nonumber \\ &= & \mathbb{P} \left[ P_{a} (R_a^2 + h_a^2)^{-{\eta_{a,L}}/2} > P_{{\rm g}} {(R_{\rm g}^2+ h_{\rm g}^2)}^{-\eta_{\rm g}/2}, P_L(R_a) \geq \frac{1}{2} \right]\nonumber \\ &=& \mathbb{E}_{R_a} \left[ \mathbb{P}\left[ P_{a} (r_a^2 + h_a^2)^{-{\eta_{a,L}}/2} > P_{{\rm g}} {(R_{\rm g}^2+ h_{\rm g}^2)}^{-\eta_{\rm g}/2}, P_L(r_a) \geq \frac{1}{2} \right] \right] \nonumber \\ &=& \mathbb{E}_{R_a} \left[ \mathbb{P}\left[ R_g^2 > \left(\frac{P_{g}}{P_a} \right)^{2/\eta_{\rm g}} (r_a^2+h_a^2)^{\eta_{a, L}/\eta_{g}} - h_{\rm g}^2 , P_L(r_a) \geq \frac{1}{2} \right] \right] \tag{5} \end{eqnarray} I'm not sure how to proceed from here. Since we are conditioning on $R_a$, $P_L(r_a) \geq \frac{1}{2}$ is now a deterministic event. I don't know how to deal with it. Any help?
Update
Let $R_g^2 > \left(\frac{P_{g}}{P_a} \right)^{2/\eta_{\rm g}} (r_a^2+h_a^2)^{\eta_{a, L}/\eta_{g}} - h_{\rm g}^2$ be event $A$ and $P_L(r_a) \geq \frac{1}{2}$ be event $B$. Conditioning on $R_a$ makes $B$ deterministic, i.e., $\mathbb{P}[B]$ is equal to $1$. Also, when we condition, events $A$ and $B$ become independent of each other. This we can write \begin{eqnarray} \mathbb{P}[A, B]&=&\mathbb{E}_{R_a}[\mathbb{P}[A, B]] \nonumber \\ &=& \mathbb{E}_{R_a}[\mathbb{P}[A] \mathbb{P}[B]] \nonumber \\ &=& \mathbb{E}_{R_a}[\mathbb{P}[A]] \nonumber \\ &=& \int_0^\infty \mathbb{P}[A] f_{R_a}(r_a) \mathrm{d}r_a\nonumber \\ &=& \int_0^\infty \mathbb{P}[R_g^2 > \left(\frac{P_{g}}{P_a} \right)^{2/\eta_{\rm g}} (r_a^2+h_a^2)^{\eta_{a, L}/\eta_{g}} - h_{\rm g}^2] f_{R_a}(r_a) \mathrm{d}r_a\nonumber \\ &=& \int_0^{\alpha_a} f_{R_a}(r_a) \mathrm{d}r_a + \int_{\alpha_a}^\infty \mathbb{P}[R_g > \sqrt{f_a(r_a)}] f_{R_a}(r_a) \mathrm{d}r_a\nonumber \\ &=& F_{R_a}(r_{a}) + \int_{\alpha_a}^{\infty} [1-F_{R_{g}}(\sqrt{f_a(r_a)})] \, f_{R_a}(r_{a}) \, dr_{a} \nonumber \\ &=& 1- \int_{\alpha_a}^{\infty} F_{R_{g}}(\sqrt{f_a(r_a)}) \, f_{R_a}(r_{a}) \, dr_{a} \nonumber \end{eqnarray} where $\alpha_a$ is a non-negative root of $f_a(r_a)$ that is given by $$ \alpha_a= \begin{cases} f{ g}(0), & f_{g}(0) >0 \\ 0, & {\rm otherwise} \end{cases} $$ then $$ p_a= \frac{1- \int_{\alpha_a}^{\infty} F_{R_{g}}(\sqrt{f_a(r_a)}) \, f_{R_a}(r_{a}) \, dr_{a}}{F_{R_a}\left( \frac{h_a}{\tan\left(\frac{\pi}{180}\left[a+ \frac{\ln(a)}{b}\right]\right)} \right)} $$ I ran a simulation of the scenario with a certain set of parameters and got $p_a=0.37$, while the evaluation of the expression gave me $p_a=0.35$. What is wrong with my analysis?