Conditional Mean/Variance Help 
Suppose you are deciding between three different transportation options, each with an average traveling time of $\mu_j$ hours and a standard deviation $\sigma_j$ hours. You randomly choose between the three options with equal probabilities. Let $T$ be the length of your trip. $\newcommand{\Var}{\operatorname{Var}}$Find $\operatorname E(T)$ and $\Var(T)$.

So, your expected travel time is simply $\operatorname E(T) = (\mu_1 + \mu_2 + \mu_3) \cdot \frac{1}{3}.$
However, I'm a bit stuck on finding the variance. I understand you use the following formula, but don't know the interior values: $\Var(T) = \operatorname E(\Var(T\mid j)) + \Var(\operatorname E(T\mid j))$.
$\Var(T\mid j)$, I believe, should be $\sigma_1^2 + \sigma_2^2 + \sigma_3^2$, but am stuck moving forward. any help?

Here, let $J$ be the random choice of transportation. Then, $\text{Var}(T \mid J)$ and $E[T \mid J]$ represent the variance and expectation of $T$ given you have chosen the $J$th transportation option and thus ought to depend on the value of $J$.
Can you compute these two "interior" values?

 $E[T \mid J=j] = \mu_j$ and $\text{Var}(T \mid J=j) = \sigma_j^2$.

Then, $E[\text{Var}(T \mid J)]$ is easy to compute, since it is $\frac{1}{3} \text{Var}[T \mid J=1] + \frac{1}{3} \text{Var}[T \mid J=2] +\frac{1}{3} \text{Var}[T \mid J=3]$.
For the variance term, just think of how you would compute the variance of a random variable that takes the values $E[T \mid J=1]$, $E[T \mid J=2]$, $E[T \mid J=3]$ with equal probability.
