## At The Home Field

There was a neat little fluke in baseball the other day. All fifteen of the Major League Baseball games on Tuesday were won by the home team. This appears to be the first time it’s happened since the league expanded to thirty teams in 1998. As best as the Elias Sports Bureau can work out, the last time every game was won by the home team was on the 23rd of May, 1914, when all four games in each of the National League, American League, and Federal League were home-team wins.

This produced talk about the home field advantage never having it so good, naturally. Also at least one article claimed the odds of fifteen home-team wins were one in 32,768. I can’t find that article now that I need it; please just trust me that it existed.

The thing is this claim is correct, if you assume there is no home-field advantage. That is, if you suppose the home team has exactly one chance in two of winning, then the chance of fifteen home teams winning is one-half raised to the fifteenth power. And that is one in 32,768.

This also assumes the games are independent, that is, that the outcome of one has no effect on the outcome of another. This seems likely, at least as long as we’re far enough away from the end of the season. In a pennant race a team might credibly relax once another game decided whether they had secured a position in the postseason. That might affect whether they win the game under way. Whether results are independent is always important for a probability question.

But stadium designers and the groundskeeping crew would not be doing their job if the home team had an equal chance of winning as the visiting team does. It’s been understood since the early days of organized professional baseball that the state of the field can offer advantages to the team that plays most of its games there.

Jack Jones, at Betfirm.com, estimated that for the five seasons from 2010 to 2014, the home team won about 53.7 percent of all games. Suppose we take this as accurate and representative of the home field advantage in general. Then the chance of fifteen home-team wins is 0.537 raised to the fifteenth power. That is approximately one divided by 11,230.

That’s a good bit more probable than the one in 32,768 you’d expect from the home team having exactly a 50 percent chance of winning. I think that’s a dramatic difference considering the home team wins a bit less than four percent more often than 50-50.

The follow-up question and one that’s good for a probability homework would be to work out what are the odds that we’d see one day with fifteen home-team wins in the mere eighteen years since it became possible.