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Week of Feb. 28, 2004; Vol. 165, No. 9 , p. 131

Toss Out the Toss-Up: Bias in heads-or-tails

Erica Klarreich

If you want to decide which football team takes the ball first or who gets the larger piece of cake, the fairest thing is to toss a coin, right? Not necessarily.

A new mathematical analysis suggests that coin tossing is inherently biased: A coin is more likely to land on the same face it started out on.

"I don't care how vigorously you throw it, you can't toss a coin fairly," says Persi Diaconis, a statistician at Stanford University who performed the study with Susan Holmes of Stanford and Richard Montgomery of the University of California, Santa Cruz.

In 1986, mathematician Joseph Keller, now an emeritus professor at Stanford, proved that one fair way to toss a coin is to throw it so that it spins perfectly around a horizontal axis through the coin's center.

Such a perfect toss would require superhuman precision. Every other possible toss is biased, according to an analysis described on Feb. 14 in Seattle at the annual meeting of the American Association for the Advancement of Science.

The researchers' logic goes like this. At the opposite extreme from Keller's perfect toss is a completely biased toss, in which the coin stays flat while in the air. Since the coin never actually flips, it is guaranteed to land on the same face that it started out on.

Between the perfectly spinning toss and the flat toss lies a continuum of other possibilities, in which the coin spins around a tilted axis, precessing like an old-fashioned children's top. Each of these possibilities is biased, the team found. The bias is most pronounced when the flip is close to being a flat toss. For a wide range of possible spins, the coin never flips at all, the team proved.

In experiments, the researchers were surprised to find that it's difficult to tell from watching a coin whether it has flipped. A coin toss typically takes just half a second, with the circumference of the coin whizzing around at 3 meters per second. What's more, the coin's spin makes it wobble, often creating the illusion that the coin has flipped.

"Sometimes we had the complete impression that the coin had turned over when it really hadn't," Holmes says.

Magicians and charlatans may take advantage of this illusion. Keller observes, "Some people can throw the coin up so that it just wobbles but looks to the observer as if it is turning over." To see whether the predicted bias shows up in actual coin tosses, the team made movies of tossed coins and then calculated the axes of spin.

Their preliminary data suggest that a coin will land the same way it started about 51 percent of the time. It would take about 10,000 tosses before a casual observer would become aware of such a small bias, Diaconis says. "Maybe that's why society hasn't noticed this before," he says.

This slight bias pales when compared with that of spinning a coin on its edge. A spinning penny will land as tails about

80 percent of the time, Diaconis says, because the extra material on the head side shifts the center of mass slightly.

During World War II, South African mathematician John Kerrich carried out 10,000 coin tosses while interned in a German prison camp. However, he didn't record which side the coin started on, so he couldn't have discovered the kind of bias the new analysis brings out.

Says David Aldous, a statistician at the University of California, Berkeley, "This is a good lesson that even in simple things that people take for granted, there may be unexpected subtleties."


Your article describes a great theory in a theoretical world. The purpose of a coin toss is to determine an outcome in the real world, however. Did the guys doing the various analyses factor in the effect of the coin bouncing on the ground or being fumbled in an attempted catch?

Ed Eierman
Romney, W.Va.

The study assumes that the toss is caught cleanly in the thrower's hand. Fumbling the catch or letting the coin bounce on the floor would change the physics drastically (and complicate the relevant equations enormously) and could indeed affect the extent to which the toss is biased.—E. Klarreich

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Diaconis, P. 2004. The search for randomness. American Association for the Advancement of Science annual meeting. Feb. 14. Seattle.

Further Readings:

Keller, J.B. 1986. The probability of heads. American Mathematical Monthly 93(March):191-197.

Peterson, I. 2004. Heads or tails? Science News Online (Feb. 28). Available at http://www.sciencenews.org/articles/20040228/mathtrek.asp.


David Aldous
Department of Statistics
University of California, Berkeley
367 Evans Hall
Berkeley, CA 94720-3860

Persi Diaconis
Department of Statistics
Stanford University
Stanford, CA 94305-4065

Susan P. Holmes
Department of Statistics
Stanford University
Stanford, CA 94305-4065

Joseph B. Keller
Stanford University
Department of Mathematics
Stanford, CA 94305-2125

Richard Mongomery
Department of Mathematics
University of California , Santa Cruz
1156 High Street
Santa Cruz, CA 95064

From Science NewsVol. 165, No. 9, Feb. 28, 2004, p. 131.

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