I recently found a new U.S. penny in my loose change. The "heads" side still shows Abraham Lincoln's profile, but the "tails" side now features the Union Shield instead of the Lincoln Memorial. I had to wonder whether this new coin has the same distinctive bias that older Lincoln pennies display.
Starting in 2010, U.S. pennies featured the Union Shield instead of the Lincoln Memorial on one side of the coin.
Flipping a coin in the air is a common way to make a choice between two alternatives; for example, to decide which team starts with the ball in a football game. You expect the coin to come up heads or tails with equal probability.
But this happens only when a coin spins perfectly around a horizontal axis through the coin's center. Everyday coin tosses typically fall short of such perfection. For these imperfect tosses, Persi Diaconis, Susan Holmes, and Richard Montgomery showed that a coin is slightly more likely (51 percent) to land on the same face as it started out on (see "Heads or Tails?").
A coin that rolls along the ground or across a table after a toss introduces other opportunities for bias. An uneven distribution of mass between the two sides of a coin and the nature of its edge can tilt the outcome to favor one side over the other.
Older U.S. pennies, showing the Lincoln Memorial, provide a striking example of such a bias. Stand a dozen or more pennies on edge on the surface of a table. 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.
A U.S. Lincoln penny standing on edge.
You might expect about half to be heads and half to be tails. But that rarely happens. Nearly always, you end up with many more heads than tails.
Such an outcome points to the fact that this coin has an uneven weight distribution, leading to strongly biased results.
Spinning a coin on its edge on a table is a somewhat different matter. Again, the location of the coin's center of gravity makes a difference, but spun pennies, for example, tend to land tails more often than heads. Indeed, a spinning penny will land tails about 80 percent of the time.
Such unexpected results come as a surprise to most people, who rarely consider the possibility that the coins they use to play games or settle a variety of issues may be biased. To ensure a more equitable result, it's probably wise to catch a coin before it lands on some surface and rolls, spins, or bounces to a stop.
I don't know yet whether the shield penny also shows a bias. I haven't managed to collect enough of the new coins to try the same experiment with them.
Are other coins biased? The best way to find out is by doing the penny experiment with them, if you can get them to stand upright.
Photos by I. Peterson