DYNAMICAL BIAS IN THE COIN TOSS
Persi Diaconis Susan Holmes Richard Montgomery
Departments of Mathematics Department of Statistics Department of Mathematics
and Statistics Sequoia Hall University of California
Stanford University Stanford University Santa Cruz
We analyze the natural process of flipping a coin which is caught in the hand. We
prove that vigorously-flipped coins are biased to come up the same way they started.
The amount of bias depends on a single parameter, the angle between the normal to
the coin and the angular momentum vector. Measurements of this parameter based
on high-speed photography are reported. For natural flips, the chance of coming up as
started is about .51.
Coin-tossing is a basic example of a random phenomenon. However, naturally tossed
coins obey the laws of mechanics (we neglect air resistance) and their flight is determined
by their initial conditions. Figure 1 a-d shows a coin-tossing machine. The coin is placed on
a spring, the spring released by a ratchet, the coin flips up doing a natural spin and lands
in the cup. With careful adjustment, the coin started heads up always lands heads up â€“ one
hundred percent of the time. We conclude that coin-tossing is â€˜physicsâ€™ not â€˜randomâ€™.
Figure 1.a Figure 1.b
Figure 1.c Figure 1.d
Joe Keller [Keller, 1986] carried out a study of the physics assuming that the coin spins
about an axis through its plane. Then, the initial upward velocity and the rate of spin de-
termine the final outcome. Keller showed that in the limit of large initial velocity and large
rate of spin, a vigorous flip, caught in the hand without bouncing, lands heads half the time.
This work is described more carefully in Section Two which contains a literature review of
previous work on tossed and spinning coins.
The present paper takes precession into account. Real flips often precess a fair amount
and this changes the conclusion. Consider first a coin starting heads up and hit exactly in
the center so it goes up without turning like a