A Penny Surprise

Flipping a coin in the air, catching it, then determining whether it has come up heads or tails is to many people a prototypical random process. In some sense, however, coin tossing isn't really random at all. A mechanical gadget can flip a properly positioned coin so that the coin always lands showing the same face.

The efficacy of such a device depends on the fact that a tossed coin obeys Newton's laws of motion. The impulse given to the coin and the distance traveled decide the result of each flip. Precisely controlling the impulse would allow you to predict how the coin would fall. Any randomness would be not in the flipping itself but in the precision with which the starting conditions were known.

In the physics of coin flipping, the most important parameters are the coin's upward velocity and its rate of spin. When the spin rate is low, the coin acts like a tossed pizza. It's unlikely to turn over more than a few times, even if its upward velocity is very large and it travels a long distance. A coin may also come down without flipping over if it doesn't go high enough. There simply could be too little time for the coin to turn over.

By calculating how often a coin turns over for a certain spin and upward velocity, you can predict whether it will come up heads or tails. A graph plotted to show the outcomes of various spin and velocity combinations reveals that for spins and velocities typically encountered in coin tosses, tiny changes in initial conditions make the difference between heads and tails.

Why then is the outcome of a coin toss considered to be random, even though it is uniquely determined by the laws of physics and the initial conditions?

Joe Keller of Stanford University has shown that for large values of the initial velocity, the sets of initial velocity values that lead either to heads or to tails are of equal size for a fair coin. Thus, half of the initial conditions lead to heads and half to tails.

In general, the size of that fraction is determined by the mechanics of the device -- whether it is a coin, a roulette wheel, or a die, and the probability of a given outcome is independent of the probability distribution of the initial conditions.

Experimental data suggest that a well-tossed fair (or unbiased) coin is a satisfactory randomizer for achieving an equal balance between two possible outcomes. However, this equity of outcome doesn't necessarily apply to a coin that rolls along the ground or across a table after a toss. An uneven distribution of mass between the two sides of a coin and the nature of its edge can slightly bias the outcome to favor, say, heads over tails.

The U.S. penny -- with Abraham Lincoln's head on one side and the Lincoln Memorial on the other -- provides a striking example of such a bias. Stand a dozen or so pennies on edge on the surface of a table. Then bang the table so that the pennies topple over. You'll find that nearly always more heads than tails are face up. Sometimes all the coins end up heads. On the other hand, spinning pennies tend to land tails more often than heads!

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 an equitable result, it's probably wise to catch a penny before it lands on some surface and rolls, spins, or bounces to a stop.

I'm not sure whether coins other than the U.S. Lincoln penny show a similar bias. I've heard, for example, that Canadian coins are minted with designs that are supposed to produce evenly balanced coins. It would be interesting to experiment with various coins to see if it's possible to detect a bias.

References:

Beasley, John D. 1990. The Mathematics of Games. Oxford, England: Oxford University Press.

Davis, Paul. 1997. Joe Keller: Card games and races, models and theories. SIAM News 30(December):4.

McKean, Kevin. 1987. The orderly pursuit of pure disorder. Discover (January):72-81.

Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

Vulovic, Vladimir Z., and Richard E. Prange. 1986. Randomness of a true coin toss. Physical Review A 33(January):576-582.

Read about the "Great International Pennies Toss" at

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.