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 perspective…