February 5, 2011

Penny Bias

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 "Coin Toss Randomness").

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 mass 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.


John Cook said...

Andrew Gelman and Deborah Nolan wrote a paper entitled You can load a die, but you can't bias a coin. It's surprisingly hard to bias a coin toss, even if you stick putty on one side of the coin. It's much easier to bias a spinning coin.

Anonymous said...

Coins are die struck and pennies have a flat edge. I suspect the bias observed was less due to unequal weight but to a slight angle of the edge done intentionally to remove the coin from the die. This might explain the opposite bias of spinning and balancing. The spin would be about the rotational center of gravity with the wider edge causing a precession. The balance would be on the flat but slightly tilted the opposite way.

Carol Covin said...

I have three questions. Is it true that the Lincoln side of a penny is heavier? If true, is it also true that when tossing a coin, the Lincoln side lands down more often? And, finally, if true, why would the Lincoln side land up more often when you spin the penny? Thanks for your help.

Anonymous said...

i concluded that a penny is more likely to come up tails, i had tested it you should try it and see what you came up with

Anonymous said...

Do you most likely come up with heads or tails because i had tested it 30 time si fliped a penny and i most likely came up with tails

Anonymous said...

in my opinion that's not true

Barbara Degnan said...

My son just completed a similar experiment for a 2nd grade project. He stood 21 pennies on their sides, then tapped the table to cause them to fall. He completed this 20 times. His results show a strong bias toward a penny on it's side falling so that it lands heads up. He used only pennies with the Lincoln Memorial on the "tails" side and also removed those that had any "gunk" stuck to one side or the other which could have changed the weight and mass. Out os the 20 trials, the smallest number of heads was 15 and the largest was all 20.

LSquared32 said...

The bias in the old pennies (stood on edge) is due to the beveling of the edges. You can measure this by taking 20 pennies and lining them up with alternating heads-tails and measuring the length of the string, and taking another 20 pennies and lining them up with all heads up and measuring the length of the string. The heads-tails string is shorter because the bevels are opposite. Now I need some new shield pennies to experiment with!