The toss that is random without chaos

The error that doubles on a schedule opens the study of chaos with two double pendulums released a hundred-millionth of a degree apart. For ten seconds they move together, and then they have nothing to do with each other, because the difference between them was growing exponentially the whole time. That is the property that defines chaos — a positive Lyapunov exponent — and it is a common explanation of why things are unpredictable. Weather is chaotic, the argument goes, and so are dice and roulette wheels and coin tosses, and that is why they are random.

The coin toss does not fit. Its motion is among the simplest in mechanics, the free flight of a spinning rigid disc, and it is not chaotic at all. Two tosses that start almost alike stay almost alike. Yet a coin toss is the standard of randomness, the device that settles cup finals and the textbook example of a fair trial. How a deterministic, non-chaotic system produces outcomes that pass every test of randomness, and why it nevertheless falls very slightly short of being fair, are two separate questions, and the answers to both are mechanical.

Every toss already decided #

The analysis was made by Joseph Keller in 1986. Take a coin launched upward at speed $v$ and spinning at angular velocity $\omega$ about a horizontal diameter, and catch it at the height it left the hand. It is in the air for a time $2v/g$, and in that time it turns through an angle $2v\omega/g$. Starting heads up, it lands heads up if it has completed an even number of half-turns and tails up if an odd number.

The result is a map of the plane of launch conditions, carved into bands by the hyperbolae $v\omega = n\pi g/2$, one for each half-turn, alternating between the starting face and the other. Every point on the map is a definite outcome. Nothing about the coin’s flight is random, and the boundaries between heads and tails are smooth curves. There is none of the fractal intermingling of outcomes that marks a chaotic system, in which arbitrarily close to any launch there are launches with the other result. Close to any launch here there are only launches with the same result, up to a definite distance, and the distance is the width of a band.

Near the origin the bands are wide. A slow, low toss with little spin has few half-turns in it and can be aimed; a magician who wishes to toss a coin honestly-looking and land it as desired works in that corner, with a toss that wobbles convincingly but does not truly flip. The bands narrow as the speed and spin grow, because the number of half-turns grows as their product, and a real toss with the thumb puts the coin far out on the map.

How much a hand has to wobble #

A typical toss, measured with high-speed cameras, leaves the thumb at about two and a half metres a second, spinning at about thirty-eight revolutions a second, and stays in the air for half a second. It completes about thirty-seven half-turns.

At that launch, the bands are 6.4 radians a second wide in spin — 2.7 per cent of the spin itself. A hand that could repeat its spin and speed to within one per cent every time could bias the outcome, but only partially: with a one-per-cent scatter in each, the chance of the starting face falls from certainty to about a third. With a two-per-cent scatter it is already within a hundredth of a half. Human tossers, measured, scatter by several per cent.

So the randomness of a coin toss is located precisely, and it is not in the coin. It is in the hand. The flight is a deterministic map from launch conditions to outcome, and what makes the outcome unpredictable is that the launch conditions are themselves uncertain, by more than the width of the bands they fall in. The coin magnifies that uncertainty only in the sense that it has many half-turns, so that a small fractional change in spin is a large change in the number of half-turns. It does not magnify it exponentially. Doubling the precision of the hand halves the scatter of half-turns; it does not buy a fixed extra fraction of a second of predictability, as it would for a chaotic system.

The same structure underlies a good many devices that are random by design. A roulette wheel has a ball that spins down in a nearly deterministic way and a mechanism of deflectors and pockets that adds a little genuine chaos at the end; physicists in the 1970s built shoe-mounted computers that timed the ball and the wheel and predicted the octant of the result well enough to beat the house edge. A die thrown onto a table has its outcome decided mostly by the bounces, which are closer to chaotic. The coin is the purest case: all of its randomness is inherited.

The difference that only adds up #

The distinction between linear and exponential sensitivity is the whole difference between a coin and a pendulum, and it is worth seeing directly.

Two coins launched together, with spins differing by a thousandth, drift apart in orientation at a constant rate: after a time $t$ their difference is $\delta\omega\, t$. When they are caught, half a second later, they differ by an eighth of a radian, and they show the same face. Two states of a chaotic system starting a similar small distance apart grow apart as $e^{\lambda t}$, and with a modest Lyapunov exponent of eight per second they pass half a turn’s worth of separation in a fraction of the coin’s flight. On the logarithmic plot the chaotic difference is a straight line climbing steeply; the coin’s difference is a curve that flattens, since a linear growth is logarithmic on such a scale.

The coin’s motion is integrable in the technical sense that the last curve to go explains for the solar system: it has as many conserved quantities as it has degrees of freedom — energy, the three components of angular momentum, and its rotational symmetry — and its motion in phase space is confined to smooth tori on which nearby trajectories drift apart only linearly. Free rigid bodies are integrable; it is what makes the axis that will not hold so clean an experiment, where a book spun about its middle axis flips over and over in a perfectly repeating way. A coin is a very symmetric rigid body and its free flight is about as integrable as anything gets.

The distinction matters because the two kinds of unpredictability respond differently to better measurement. For a chaotic system, each tenfold improvement in knowing the starting state buys a fixed additional time of prediction — one more doubling time per factor of two. For a coin, a tenfold improvement in knowing the launch makes the outcome tenfold more predictable at a fixed number of half-turns, and a machine that launches coins with high precision can make them land the same way every time. Persi Diaconis, Susan Holmes and Richard Montgomery built one: a spring-loaded launcher whose coins came up heads every time they were loaded heads up.

The face that stays up longer #

Keller’s analysis assumes the coin spins about an exact diameter, so that its face normal turns in a vertical plane and spends exactly half its time pointing up. Real tosses are not so tidy. The thumb flicks one edge of the coin, and the angular momentum it imparts is not exactly in the plane of the coin. A rigid disc whose angular momentum is tilted from its plane does not simply spin; its normal precesses around the angular momentum, sweeping out a cone — the same free precession that makes a badly thrown rugby ball wobble as it flies.

The geometry is simple once the angle is named. Let $\psi$ be the angle between the angular momentum and the coin’s normal. The normal circles the angular momentum at that fixed angle, and if it starts pointing straight up, its vertical component at a later moment is $\cos^2\psi + \sin^2\psi\cos\omega t$. For a coin spun exactly about a diameter, $\psi$ is ninety degrees and the vertical component swings symmetrically between plus and minus one. For any smaller tilt it swings between one and $\cos 2\psi$, which is greater than minus one, and so the starting face spends more of its time pointing up than down. At a tilt of seventy degrees it is up 54 per cent of the time; at fifty degrees, 75 per cent; below forty-five degrees it never turns over at all, and the coin, which looks to be flipping vigorously, is only wobbling. A conjuror’s “honest” toss uses exactly that.

The bias a wobble gives a fair coin #

Diaconis, Holmes and Montgomery worked out the consequence in 2007. A coin caught at a time that is effectively random with respect to its precession shows its starting face with probability

$$P = \frac{1}{2} + \frac{1}{\pi}\arcsin(\cot^2\psi),$$

for tilts between forty-five and ninety degrees, and with certainty below forty-five.

Their high-speed films of people tossing coins showed tilts typically around eighty degrees, which the formula turns into a probability of about fifty-one per cent for the face that started up. The effect is small and the prediction was, for some years, an unconfirmed curiosity, since detecting a one-per-cent bias needs tens of thousands of tosses for a decisive result. In 2023 František Bartoš and forty-seven colleagues tossed 350,757 coins, recording which face was up at the start and which at the end, and found the starting face came up 50.8 per cent of the time. The prediction, from the mechanics of a precessing disc, was correct to the precision of the count.

The bias does not depend on whether the coin is heads or tails up; it depends only on which face started up. Anyone who does not know how the coin lies in the tosser’s hand before the toss still faces a fair bet. The bias also varied between tossers in the experiment, from almost none to several per cent, as the formula says it should: it is a property of how a person throws — of their personal tilt — and not of the coin.

What a coin says about probability #

The coin toss is often presented as the model of a random experiment, and the mechanics shows what that model rests on. The probability of a half is not a property of the coin’s flight, which is deterministic. It is a statement about the launch conditions: that they are spread over many bands of the map in roughly equal measure. Physicists call this “the method of arbitrary functions”, after Henri Poincaré, who noticed in the 1890s that any reasonably smooth distribution of launch conditions spread over many narrow bands produces nearly equal probabilities for the alternating outcomes, whatever the distribution’s shape. The coin’s fairness is then a theorem about smooth distributions on finely banded maps.

It is the same structure that underlies statistical mechanics, which entropy is a count builds on counting the microscopic arrangements compatible with a macroscopic state. The microscopic motion of a gas is deterministic, and the probabilities of statistical mechanics are statements about how initial conditions are spread over the regions of phase space that lead to each outcome. Gases are strongly chaotic, which mixes those regions finely and quickly; a coin achieves its fine banding with no chaos at all, by simply turning many times. Chaos is one way of turning a smooth spread of starting conditions into equal probabilities; it is not the only one.

The precession bias then says something precise about when such arguments fail. The banded map assumes the outcome depends on a single number, the count of half-turns. Once the coin precesses, the outcome depends also on the tilt, and the tilt is not spread over many bands: it is concentrated near a typical value set by how people throw. The fine-banding argument applies to the spin and fails for the tilt, and the residual bias is the trace of the one launch variable that is not finely mixed.

Spinning, bouncing and other ways to lose fairness #

Coins are not always tossed and caught. A coin spun on its edge on a table and allowed to fall, rather than flipped, behaves very differently: its outcome is decided by its slight asymmetries of weight and shape over the long time it spins, and coins of some designs land heads or tails far from half the time, sometimes by tens of per cent. A coin allowed to land on a hard floor and bounce adds collisions to the flight, and the bounces, which are much closer to chaotic, scramble the outcome further and reduce the precession bias, since a bouncing coin forgets how it started more thoroughly than a caught one. The 50.8 per cent applies to the caught toss that Diaconis, Holmes and Montgomery analysed, and to the experiment that tested it.

Where chaos does enter #

Other random devices are less pure than the coin, and where they gain their randomness is instructive. A rigid body tumbling freely is integrable, but one that is pushed by something else need not be: the rod that tumbles for ever in a stream shows a slender body turning periodically in a shear flow, regular because the flow is simple, and a body driven by a more complicated flow can tumble chaotically. A die thrown onto a table bounces several times, and each bounce is a collision whose outcome depends sensitively on the orientation at impact; a sequence of such collisions is the stretching and folding that the fold that has to be there identifies as the engine of chaos. A die’s outcome is decided mostly by its last few bounces, where that amplification has had time to act, rather than by the throw.

Even then the amplification is limited by the number of bounces. A die that bounces four times before settling has been through four stretchings, and its sensitivity to the throw, though much greater than a coin’s, is finite. Experiments with dice thrown by machines find that the outcome is predictable from the throw when the bounces are few, and becomes unpredictable only as they multiply — which is why casinos require dice to be thrown against a textured wall. The randomness of dice is a mixture: inherited from the hand like the coin’s, and multiplied by a chaotic stage that a coin, caught in the air, never reaches.

And chaos, where it does act, has its own subtlety. The wrong orbit beside a right one shows that a computed trajectory of a chaotic system, though wrong in detail after a few steps, stays close to some true trajectory of the same system for a long time. A chaotic device does not create outcomes from nothing; it maps a spread of starting conditions onto a spread of outcomes, as the coin does, only with a map whose bands become exponentially finer with every step. The difference between the coin and the die is the rate at which the bands narrow, not whether they exist. A dripping tap, which the map a dripping tap turns out to be reduces to a rule applied once per drop, narrows its bands by a fixed factor at every drip; a coin narrows them only by flipping more times.

A thin disc caught where it left #

The map of bands assumes a coin spinning exactly about a horizontal diameter and caught exactly at its launch height; catching it lower or higher shifts every band, and catching it in a palm that turns it over reverses the outcome, without changing the structure. The simulated scatter is Gaussian and equal in speed and spin, an illustration rather than a measured distribution of human tosses. The chaotic comparison uses an exponent of eight per second, chosen to be of the order of a double pendulum’s, to show the shapes of the two kinds of growth rather than to model any particular system. The precession formula treats the coin as a thin uniform disc with no air resistance, with the catch at a random phase of the precession; real coins have a small air drag on their spin and a thickness that slightly changes their moments of inertia. The tilt that gives 50.8 per cent is read off the formula, not measured.

Still open: how much of the randomness of the world is inherited #

The coin is a clean case in which randomness is entirely inherited from the uncertain conditions of a deterministic process, with no chaos in between. How much of the apparent randomness in physics is of that kind, and how much is generated by chaotic amplification of microscopic fluctuations — perhaps ultimately quantum ones — is a question with no simple answer. For a coin, a back-of-envelope estimate suggests that the thermal and quantum jitter of the thumb’s muscles is far too small to matter; the randomness is macroscopic. For the weather, chaotic amplification from tiny disturbances is the accepted story. For many processes in between, such as dice, roulette and the dynamics of granular materials, the balance between inheritance and amplification has been estimated in particular cases but not settled in general.

The habit worth carrying away is to ask where the randomness comes from. A coin’s flight is integrable and its sensitivity linear, so its outcome is a smooth map of launch speed and spin in bands only a few per cent wide, and the toss is fair because a human hand scatters by more than that — not quite fair, because the coin precesses at a tilt near 81° and keeps its starting face up 50.8 per cent of the time, as 350,757 tosses confirmed. Chance, in a coin, is a hand’s imprecision measured against a stripe.