Paulos: Statistical Ties and Coin Flips

The term refers to the way a butterfly flapping its wings in Central America can lead to a snowstorm in New York a few months later. More generally, any small change occurring in some quantity can cascade toward a hugely disproportionate consequence down the road. In this case the consequence is the identity of the next president of the United States.

Rather than examining other disputed vote totals mathematically, I’ll end by recalling a classic problem in the history of probability that is relevant to the problem of what to do in indeterminate situations.

Proceeding Under Uncertainty

Consider this scenario: Two men, A and B, bet on a series of coin flips. They agree that the first one to win 6 such flips will be awarded $64,000. The game, however, is interrupted after only 8 flips with A leading B 5 to 3. The question is how should the $64,000 pot be divided?

One argument might be that A should be awarded the full $64,000 since the bet ought to be all or nothing and he was leading. Note the non-mathematical “ought” here.

Another tack might be to maintain instead that A should receive 5/8 of the pot, $40,000, and B the remaining 3/8, $24,000, since the score was 5 to 3 when the coin flips ended. Proportional division has a long tradition.

One could also reason that neither A nor B won the game so the pot ought to be split evenly with each receiving $32,000. Another “ought” with a long history.

Yet another approach involves the probability that A or B would have gone on to win. The likelihood that B would have won had the flipping continued is 1/8 since he would have needed to win three consecutive flips, and the probability of this is 1/8. Thus B should receive 1/8 of the pot or $8,000. And since the likelihood that A would have won is 7/8, he should receive the remaining $56,000.

Settling on the Rules

Other divisions are possible as well. (The above problem would be further complicated if, like the role of Florida in the presidential election, the coin-flipping were part of another larger game that would result in a complete victory for B if he received any of the $64,000 pot.)

The point is the criteria for these divisions of the $64,000 pot are not mathematical!

Unfortunately in both the Florida election and this little example, the criteria for deciding the issue are not at all obvious. Flipping a coin, dividing things equally or proportionally, hiring small armies of attorneys, organizing a foot race or perhaps a vocabulary competition between the candidates — none of these options is specified in election law, which is, by turns, silent, clear, irrelevant, and contradictory.

Mathematics may help determine the consequences of our assumptions and values. But we, not some mathematical divinity, are the source of these assumptions and values.

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears on the first day of every month.

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