How to calculate the following probability $$\Pr(\gamma_2 < z | \gamma_1 \geq z),$$ where $\gamma_1 \triangleq \frac{a_1g_1}{a_2 g_2 + 1}$, $\gamma_2 = a_2 g_2$, $a_1, a_2 > 0$ and $g_i, i \in \{1, 2\}$ is an exponentially distributed random variable, i.e., $f_{g_{i}}(x) = \frac{1}{\Omega_i} \exp \left( -x/\Omega_i\right)$?
The problem I am facing here is that defining $\gamma_2 < z$ as event $B$ and $\gamma_1 \geq z$ as event $A$, these two event are dependent. Therefore, using the Baye's theorem, $$\Pr(B|A) = \dfrac{\Pr(A, B)}{\Pr(A)},$$ but I am not sure how to calculate $\Pr(A, B)$. However, for $\Pr(A)$, I know that $$\Pr(A) = \Pr(\gamma_1 \geq z) = \int_z^\infty f_{\gamma_1}(x) \mathrm dx.$$
Thanks in advance!