SIAM REVIEWc!2007 Society for Industrial and Applied MathematicsVol. 49, No. 2, pp. 211–235Dynamical Bias in the Coin Toss∗Persi Diaconis†Susan Holmes‡Richard Montgomery§Abstract. We analyze the natural process of flipping a coin which is caught in the hand. We showthat vigorously flipped coins tend to come up the same way they started. The limitingchance of coming up this way depends on a single parameter, the angle between the normalto the coin and the angular momentum vector. Measurements of this parameter based onhigh-speed photography are reported. For natural flips, the chance of coming up as startedis about .51.Key words. Berry phase, randomness, precession, image analysisAMS subject classifications. 62A01, 70B10, 60A99DOI. 10.1137/S00361445044464361. Introduction. Coin tossing is a basic example of a random phenomenon. How-ever, naturally tossed coins obey the laws of mechanics (we neglect air resistance) andtheir flight is determined by their initial conditions. Figures 1(a)–(d) show a cointossing machine. The coin is placed on a spring, the spring is released by a ratchet,and the coin flips up doing a natural spin and lands in the cup. With careful adjust-ment, the coin started heads up always lands heads up—one hundred p ercent of thetime. We conclude that coin tossing is “physics” not “random.”Joe Keller [20] carried out a study of the physics assuming that the coin spinsabout an axis through its plane. Then, the initial upward velocity and the rate of spindetermine the final outcome. Keller showed that in the limit of large initial velocityand large rate of spin, a vigorous flip, caught in the hand without bouncing, landsheads up half the time. This work is described more carefully in section 2 whichcontains a literature review of previous work on tossed and spinning coins.The present paper takes precession into account. We will use the term precessionto indicate that the direction of the axis of rotation changes as the coin goes throughits trajectory (see Figure 2(a)). Real flips often precess a fair amount and this changesthe conclusion. Consider first a coin starting heads up and hit exactly in the centerso it go es up without turning like a spinning pizza. We call such a flip a “total cheatcoin,” because it always comes up the way it started. For such a toss, the angularmomentum vector!M lies along the normal to the coin, and there is no precession.∗Received by the editors October 29, 2004; accepted for publication (in revised form) September25, 2006; published electronically May 1, 2007. The first and second authors were partially supportedby NSF grant DMS 0101364.http://www.siam.org/journals/sirev/49-2/44643.html†Departments of Mathematics and Statistics, Stanford University, Stanford, CA 94305.‡Department of Statistics, Stanford University, Stanford, CA 94305 ([email protected]).§Department of Mathematics, University of California, Santa Cruz, CA 95064 ([email protected]). This author was partially supported by NSF grant DMS 0303100.211212 PERSI DIACONIS, SUSAN HOLMES, AND RICHARD MONTGOMERY(a) (b)(c) (d)Fig. 1(a) (b)Fig. 2 (a) Diagram of a precessing coin. (b) Coordinates of precessing coin:!K is the upwarddirection, !n is the normal to the coin,!M is the angular momentum vector, and ω3is therate of rotation around the normal !n.DYNAMICAL BIAS IN COIN TOSS 213In section 3 we show that the angle ψ between!M and the normal !n to thecoin stays constant. If this angle is less than 45◦, the coin never turns over. Itwobbles around and always comes up the way it started. In all of these cases there isprecession. Magicians and gamblers can carry out such controlled flips which appearvisually indistinguishable from normal flips [24].For Keller’s analysis,!M is assumed to lie in the plane of the coin with an angleof 90◦to the normal to the coin; again there is no precession. We now state ourmain theorems. The various coordinate vectors are shown in Figure 2(b). Completenotational details are in section 3.1. We use capital letters for the laboratory frameand lowercase letters for the body-centered frame. In particular, !n is the normal fromthe point of view of the coin and!N(t) is the normal from the point of view of theobserver at time t. At time zero,!N(t) =!N(0) = !n.Theorem 1. For a coin tossed starting heads up at time 0, let τ (t) =!N(t) ·!Kbe the cosine of the angle between the normal at time t and the up direction!K. Then(1.1) τ (t) = A + B cos(ωNt),with A = cos2ψ, B = sin2ψ, ωN= !!M!/I1, I1=14(mR2+13mh2) for coins with ra-dius R, thickness h, and mass m. Here ψ is the angle between the angular momentumvector!M and the normal at time t = 0, and ! · ! is the usual Euclidean norm.Theorem 1 gives a simple formula for the relevant position of the coin as a functionof the initial conditions. As shown below, the derived parameter ωNwill be large forvigorously flipped coins. To apply Theorem 1, consider any smooth probability densityg on the initial conditions (ωN, t) of Theorem 1. Keep ψ as a free parameter. Wesuppose g to be centered at (ω0, t0) so that the resulting density can be written inthe form g(ωN− ω0, t − t0). Let (ω0, t0) tend to infinity along a ray in the positiveorthant ωN> 0, t > 0, corresponding to large spin and large time-of-flight.Theorem 2. For all smooth, compactly supported densities g, as the product ω0t0tends to infinity, the limiting probability of heads p(ψ) with ψ fixed, given that headsstarts up, is(1.2) p(ψ) =!12+1πsin−1(cot2(ψ)) ifπ4< ψ < 3π/4,1 if 0 < ψ < π/4 or3π4< ψ < π.A graph of p(ψ) app ears in Figure 3. Observe that p(ψ) is always greater thanor equal to 1/2 and equals 1/2 only if ψ = π/2. In this sense, vigorously tossed coins((w0, t0) large) are biased to come up as they started, for essentially arbitrary initialdistributions g. The proof of Theorem 2 gives a quantitative rate of convergence top(ψ) as ω0and t0become large.We now explain the picture behind Theorem 1 and some heuristics for Theorem2 (see Figure 4). While the coin is in flight its angular momentum is constant intime and the normal vector precesses around it at a uniform velocity, sweeping outa circle on the sphere of unit vectors. (This is proved in section 3.) On this sphere,draw the equator of vectors orthogonal to the direction!K of “straight up.” Points onthe equator represent the coin when only its edge can be seen. Points in the upperhemisphere H represent the coin