Problem 1. Consider what happens when you throw two dice. You can ask the question "what's the probability of getting a total of 8 when the first one is 5?. Or in more generality: what's the probability of getting a total of $T$ when the first one is $m$?. The answer is a function of two variables, $T$ and $m$. You could call the result $P(m,T)$. How do you define averages in this case? For example, what's the average value of $m T$. You could determine $\langle m T \rangle$ by throwing the dice repeatedly taking the number of the first die and multiplying by the total. What you get should be equivalent to $$\langle m T \rangle = \sum_m \sum_T m T P(m,T)$$ (1.36) but that's a sum over a lot of possibilities, all possible values of $m$ and $T$. Instead use independence to solve this problem in analogy to what we just did for the two dice.