The Home-Field Paradox
We consider an apparent paradox that arises from the following puzzle. Baseball teams A and B meet annually and play until one team has won four games and is declared the series winner. The teams are evenly matched except that each team enjoys the same sl
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In Mind-Benders for the Quarantined, a challenging mathematical puzzle was—and still is, as of this writing—being offered each week by the National Museum of Mathematics (“MoMath”) to a list of subscribers that currently number about 10,000, located in 50 states and more than 80 countries. Every Sunday a new puzzle is sent out, followed by a hint on Tuesday, a bigger hint on Thursday, and a solution on Saturday. This (free) service was begun in March 2020, with the idea of supplying brain fodder for home-bound puzzle mavens during the COVID-19 pandemic. The puzzles were collected or composed by the author; the paradox discussed below arises from a Mind-Bender puzzle that he devised earlier, while the COVID19 outbreak was raging in Wuhan, for use in a MoMath course for executives called “Probability and Intuition.” The paradox concerns two baseball teams and a statistician. The teams are the Appleton Aardvarks and the Brockville Bandicoots; they are traditional rivals in their league, and meet every year to play a series of games. The first team to win four games gets to take home the coveted Cassowary Cup and display it proudly until, perhaps, the next year’s series. The teams are evenly matched except that playing on home turf confers a small advantage; in any given game between the Aardvarks and the Bandicoots, the home
This research was supported by NSF grant DMS-1600116
P. Winkler (*) National Museum of Mathematics, New York, NY, USA Department of Mathematics, Dartmouth College, Hanover, NH, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Wonders (ed.) Math in the Time of Corona, Mathematics Online First Collections, https://doi.org/10.1007/16618_2020_4
P. Winkler
team has a 54% chance of winning.1 There are no ties. By tradition, every year the first three games of the series are played in Appleton and all remaining games in Brockville. The league statistician has been looking over the records of this very longrunning series and has made a curious discovery. Intrigued, she proceeds to do some math.
1 Where Are Most of the Games Played? Every year the Cassowary Cup contest takes 4, 5, 6 or 7 games to be decided. The statistician’s intuition tells her that on average, the number of games played ought to be somewhere between 5 and 6. If so, it would follow that over the long haul, fewer than 3 games per year would be played in Brockville. Since three games are always played in Appleton, the statistician expects most series games to have been played in Appleton. But just because a number n can be 4, 5, 6 or 7 doesn’t mean these values are equally likely; the “expected value” of n, i.e., the long-run average, could be anything between 4 and 7. So the first thing our statistician does is to compute the average number of games played in the series. To keep things simple, she begins with the assumption that the outcome of every game is a 50–50 proposition. The probability that n ¼ 4 is easy to calculate; for the series to be over after only four games, the second, third, and f
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