[FoRK] Making generations of statisticians cry...

jbone at place.org jbone at place.org
Tue Mar 2 09:45:31 PST 2004


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 

"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 

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 

"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."

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