Coin flip nonrandom

Tue Mar 9 07:01:17 PST 2004

Science News Online

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

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References:

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