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In Cleveland, the Improbable becomes the Probable

July 18, 2013

"That things improbable oft will hap to men. For what is improbable does happen, and therefore it is probable that improbable things will happen."

Or so wrote Aristotle more than two thousand years ago.

While the Greek mathematician’s perspective undoubtedly applied to countless events in ancient Greece, he most certainly was not thinking about baseball, which wasn’t invented until the 19th century.


Yet, Aristotle’s words rang true this past weekend in Cleveland when fan Greg Van Niel grabbed four balls during a single game at the home-town Indians’ Progressive Field.

Van Niel’s accomplishment—or good fortune—immediately raised the question: what’s the probability of nabbing four foul balls at a major-league baseball game?

One estimate put it as a one in a trillion chance. But we decided to dig deeper using one of our own experts: Courant Professor Charles Newman, a probability expert who directed the institute from 2002 to 2006.

Image courtesy of Schyler at en.wikipedia

Here are the numbers he plugged into his calculations:

•There are nearly 50 foul balls in a major-league game

• The median major-league attendance for 2013 is approximately 30,000 fans

• There are approximately 2,400 major-league games per regular season

“Assuming all spectators are equally likely to grab each foul ball, which is a questionable assumption, someone would get four foul balls in a single game about once every 40,000 seasons,” explains Newman.

But, of course, not all fans are seated equally—those in the upper decks never come close to catching a ball headed for the seats while those sitting closer to the field may have multiple chances in a single game. Van Niel, a season-ticket holder, was sitting on the field level, down the third-base line—prime foul-ball territory.

So Newman adjusted his calculation to more realistically reflect stadium seating.

“If one replaces the bad assumption by a more reasonable one, like only about 3,000 of the spectators have any real chance of grabbing a foul,” Newman notes, “that changes the calculation by a factor of 1,000 and says it should happen about every 40 seasons—which means it may not have happened before and we'll have to wait a long time until it happens again.”

So is what happened in Cleveland last weekend highly improbable? From a mathematical standpoint, of course. But Aristotle was also a philosopher, as Newman’s Courant colleague Sylvain Cappell points out.

“Events which occur once in 40 years are, of course, individually improbable,” Cappell observes. “But, in some respects, that's really not so rare. So, comparably notable and rare events, even within the limited setting of professional baseball, are likely to, in fact, occur annually. Aristotle said it best: it's probable that in the course of life, improbable things will happen.”

Surely something for baseball fans, and especially Cleveland Indians’ fans to hold on to—their team last won the World Series in 1948, but is only 1.5 games out of a playoff spot at this year’s All-Star break.

--James Devitt

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