The Angry Statistician

As I've mentioned previously , a common method used in sports for estimating game outcomes known as log5 can be written  $$p = \frac{p_1 q_2}{p_1 q_2+q_1 p_2}$$ where $p_i$ is the fraction of games won by team $i$ and $q_i$ is the fraction of games lost by team $i$. We're assuming that there are no ties. What's the easiest way to derive this estimate? Here's one argument. Assume team $i$ has a probability $p_i$ of beating an average team (a team that wins half its games). Now imagine that this means for any given game the team has some "strength" sampled from [0,1] with median $p_i$ and that the stronger team always wins. Thus, the probability that team 1 beats team 2 is  $$p = \int_0^1 \int_0^1 \! \mathrm{Pr}(p_1 > p_2) \, \mathrm{d} p_1 \mathrm{d} p_2 .$$ This looks complicated, but but with probability $p_1$ team 1 is stronger than an average team and with probability $p_2$ team 2 is stronger than an average team. From this perspecti...