Let $T_A, T_B, T_C$ and $T_M$ are the i.i.d random variables that follows the Exponential distribution Exp(1) i.e. parameter $\lambda = 1$. I want to compute the following $$\mathbb{P}\left(\left(T_M + T_A\right) > \max \left\{T_B,T_C\right\}\right) \mid T_A < \min \left\{T_B,T_C\right\})\mathbb{P}\left(T_A < \min \left\{T_B,T_C\right\}\right)$$
I have tried to evaluate first $\mathbb{P}(T_A < \min \left\{T_B,T_C\right\})$ In this case $$\mathbb{P}(T_A < \min \left\{ T_B, T_C\right\}) \\ \mathbb{P}(T_A <T_B) \mathbb{P}(T_B < T_C)$$
After that I don't know how to proceed as two are random variable. Can some body help