Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent was approximately 2/(σ) where σ was the standard deviation of runs scored by all teams divided by the average number of runs scored. In 2006, Professor Steven J. Miller provided a statistical derivation of the formula under some assumptions about baseball games: if runs for each team follow a Weibull distribution and the runs scored and allowed per game are statistically independent, then the formula gives the probability of winning. More simply, the Pythagorean formula with exponent 2 follows immediately from two assumptions: that baseball teams win in proportion to their "quality", and that their "quality" is measured by the ratio of their runs scored to their runs allowed. For example, if Team A has scored 50 runs and allowed 40, its quality measure would be 50/40 or 1.25. The quality measure for its (collective) opponent team B, in the games played against A, would be 40/50 (since runs scored by A are runs allowed by B, and vice versa), or 0.8. If each team wins in proportion to its quality, A's probability of winning would be 1.25 / (1.25 + 0.8), which equals 502 / (502 + 402), the Pythagorean formula. The same relationship is true for any number of runs scored and allowed, as can be seen by writing the "quality" probability as [50/40] / [ 50/40 + 40/50], and clearing fractions. The assumption that one measure of the quality of a team is given by the ratio of its runs scored to allowed is both natural and plausible; this is the formula by which individual victories (games) are determined. [There are other natural and plausible candidates for team quality measures, which, assuming a "quality" model, lead to corresponding winning percentage expectation formulas that are roughly as accurate as the Pythagorean ones.] The assumption that baseball teams win in proportion to their quality is not natural, but is plausible. It is not natural because the degree to which sports contestants win in proportion to their quality is dependent on the role that chance plays in the sport. If chance plays a very large role, then even a team with much higher quality than its opponents will win only a little more often than it loses. If chance plays very little role, then a team with only slightly higher quality than its opponents will