I randomly chose either Alice or Bob to go catch a penguin for me with equal probabilities of chosing either person. Let $I=0$ if I chose Alice and $I=1$ if I chose Bob. Alice can catch a penguin in time $T_1 \sim Exponential(\lambda_1)$. Bob can catch a penguin in time $T_2 \sim Exponential(\lambda_2)$. Let $T$ be the time it takes for a penguin to be caught. What is the variance of $T$?
I started this problem with:
$$ Var(T) = E(Var(T|I)) + Var(E(T|I)) $$
However, I'm not sure of how to calculate either $E(Var(T|I))$ or $Var(E(T|I))$.