Random Baseball - The Perfect Edition

• Last Saturday White Sox Pitcher Phillip Humber pitched the 21st perfect game in MLB history. White Sox pitchers have recorded a disproportionate 3 of those games.
• Not to be out done, Andrea Chiaradio threw a perfect perfect game. She recorded all the outs in her game by strikeout. 
• Ed at Gin and Tacos pondners the rarity of perfection
• And it is rare and should be rarer. There have been over 200,000 MLB games played and only 21 of them resulted in perfect games. That is about 5 times more than the math predicts. 

  When I add this modified error rate to on-base percentage, the resulting number—.348—is the probability that the average hitter reaches base, or in other words, breaks up a perfect game. Conversely, the probability that a player does not reach base is 1 - .348 = .652. As an aside, I am treating the number of dropped third strikes and catcher’s interference calls as negligible, having no noticeable effect on the aforementioned probability. Now, to find the likelihood of a perfect game, given a historically average pitcher facing a lineup of nine historically average hitters, I multiply .652 by itself 27 times. This gives me the probability that 27 consecutive hitters will not reach base against an average pitcher. (.652)^27 = 0.00000983 As mentioned, there have been 200,304 games played in major league history, at the start of which two pitchers each have a chance to throw a perfect game. This brings us to 400,608 perfect game opportunities. Multiplying this number by the probability of the average perfect game gives me the number of perfect games that we would expect to have been thrown in history, based simply on the odds of their occurrence. 0.00000983 x 400,608 = 3.94 ≈ 4.00 perfect games, or one every 34 seasons