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MatheMUSEments
Flipping a Coin
By Ivars Peterson
Muse, April 2003, p. 19.
Heads or tails?
Flipping a coin is a common way to start off a game or settle
a question. Because you expect that the coin is as likely to come
up heads as tails, it sounds like a fair way to make a choice.
But is it really? Here's something you can try with a U.S.
penny. The results may surprise you.
Stand a dozen or more pennies on edge on the surface of a
table. It may take patience, but if the table is smooth and flat,
you should end up with an impressive cluster of upright coins.
Then let the pennies fall over. You might have to bang the table
to get them all to topple. Count how many land with tails up and
how many land with heads up.
You probably expected about half to be heads and half tails.
In fact, that rarely happens. Nearly always, you end up with far
more heads than tails.
If you look closely at the U.S. penny, you'll see the sides
aren't really flat. Abraham Lincoln's head is carved out of one
side and the Lincoln Memorial is inscribed on the other.
In fact, it looks like a little more metal has been removed from the heads
side than from the tails side. As a result, the coin tends to
fall over with the head on top. In other words, because of its
uneven distribution of weight, the penny is biased.
A penny's slight weight bias has very little effect on how it
flips in the air, however. The bias becomes important only when
the coin lands. So if you flip a penny to decide something, it's
wise to catch it before it has a chance to roll, spin, or bounce
to a stop.
Are other coins biased? The best way to find out is by doing
the penny experiment with them. Some coins won't stand upright,
but among coins that will, you'll find some are more biased than
others.
You probably knew some tricksters use a two-headed coin and
some magicians have learned how to flip a coin so it always comes
up heads. Now you know the coin itself isn't fair. One more
trusting assumption knocked on its head--or should I say on its
tail?
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