Bingo Paradox

In 29 years of teaching at Harvey Mudd College, Arthur Benjamin, the Smallwood Family Professor of Mathematics, has seen almost every possible mathematical paradox, especially ones that involve probability and games of chance.

That’s why he was astonished when a former student introduced him to the Bingo Paradox.

The Bingo Paradox, as presented in the September 2017 issue of Math Horizons, states that when playing with a large number of Bingo cards, horizontal bingos are about three times more likely than vertical bingos.

By comparison, when playing with a single card, horizontal and vertical bingos are equally likely (and more likely than a diagonal bingo). But even with just ten cards in play, it’s still the case that the winning card is more likely to be horizontal than vertical, and that edge grows as the number of cards increases.

Benjamin first learned of the Bingo Paradox from Joseph Kisenwether ’93, a former student who attended Harvey Mudd before transferring to the New Jersey Institute of Technology. Kisenwether, who develops games for the casino industry, describes his job as “the reason you can’t win.”

About five years ago, Benjamin was working on a DVD for the The Great Courses series from The Teaching Company called The Mathematics of Games and Puzzles.

“I consulted Joe because I wanted to get expert information on certain games like video poker and wanted to make sure what I said was accurate,” Benjamin said.

One lecture on the DVD was on Scams and Hustles and Kisenwether told Benjamin about the paradox, which he discovered while working for a company that published electronic bingo software. A client had complained to Kisenwether about a “bug” in the software because too often the winning card was horizontal. He determined it was not a bug but a mathematical feature.

Subsequently, Benjamin ran into Ben Weiss ’94, another mathematics alumnus who works for Google, and told him about the paradox.

“He wrote me the next day with the idea of looking at the problem with every possible card in play,” Benjamin said. The result was a collaboration among the three that resulted in the Math Horizons article.

Benjamin, is recognized nationally for his ability to perform rapid mental calculations. He has lectured and performed for audiences around the world and has published several books on how to make math both fun and easy. He is also a professional “mathemagician” and frequently performs at the Magic Castle in Hollywood and nationwide. His TED talk “Faster Than a Calculator” has generated 8 million views. And, Readers Digest calls him “America’s Best Math Whiz.” In May, Benjamin received Harvey Mudd’s Henry T. Mudd Prize for extraordinary service to the College.