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
Abstract
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.
Introduction
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 – one
hundred percent of the time. We conclude that coin-tossing is ‘physics’ not ‘random’.
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.
1
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 pizza. We call such a flip a “total cheat coin”,
because it always comes up the way it started. For such a toss, the angular momentum
vector
→
M lies along the normal to the coin.
In Section Three we prove that the angle ψ between
→
M and the normal to the coin stays
constant. If this angle is less than 45â—¦, the coin never turns over. It wobbles around and
always comes up the way it started. Magicians and gamblers can carry out such controlled
flips which appear visually indistinguishable from normal flips. For Keller’s analysis,
→
M is
assumed to lie in the plane of the coin making angle 90â—¦ with the normal to the coin.
We state our main theorems first.
Theorem 1 For a coin tossed starting heads up at time 0 the cosine of the angle between
the normal to the coin at time t and the up direction is
(1) f(t) = A+B cos(ωN t)
with A = cos2 ψ,B = sin2 ψ, ωN = ‖
→
M‖/I1, I1 = 14mR2 + 13mh2 for coins with radius R,
thickness h and mass m. Here ψ is the angle between the angular momentum vector
→
M and
the normal at time t = 0.
K
n
M
R
r
y
wN
Figure 2: Coordinates of Precessing Coin.
To apply theorem 1, consider any smooth probability density g on the initial conditions
(ωN , t) of Theorem 1. Keep ψ as a free parameter. We suppose g to be centered at (ω0, t0)
so that the resulting density can be written in the form g(ωN − ω0, t− t0). Let (ω0, t0) tend
to infinity along a ray in the positive orthant ωN > 0, t > 0, corresponding to large spin, and
large time-of-flight.
2
Theorem 2 . For all smooth, compactly supported densities g, the limiting probability of
heads p(ψ) with ψ fixed, given that heads starts up, is given by
(2). p(ψ) =
12 + 1pi sin−1(cot2(ψ)) if pi4 < ψ < 3pi/41 if 0 < ψ < pi/4 or 3pi
4
< ψ < pi.
A graph of p(ψ) appears in Figure 3. Observe that p(ψ) is always greater than or equal
to 1/2 and equals 1/2 only if ψ = pi/2. In this sense, vigorously tossed coins ((w0, t0) large)
are biased to come up as they started, for essentially arbitrary initial distributions g. The
proof of Theorem 2 gives a quantitative rate of convergence to p(ψ) as ω0 and t0 become
large.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
Figure 3: p(ψ)
We now explain the picture behind Theorem 1 and some heuristics for Theorem 2. The
angular momentum vector is constant in time and the normal vector precesses around it
at a uniform velocity, sweeping out a 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 on the equator represent the coin edge-on. Points in the upper
hemisphere H represent the coin ‘heads up’ and points in the lower hemisphere T represent
the coin ‘tails up’. H corresponds to f > 0 and T to f < 0 where f is the function of theorem
1.
Suppose now that the coin starts its travel precisely heads up – so that the normal is
aligned with
→
K. Then the normal
→
N traces out a circle on the sphere passing through
→
K
and having center the “random” point
→
M (normalized). For all choices of
→
M except for
→
M
lying in the equator (the Keller flip) more of this circle lies in the H hemisphere than the T.
The coin appears biased towards heads.
To obtain a quantitative expression for the bias we fix the angle ψ, which is also the
(spherical) radius of the circle described by the normal. To begin, note that if ψ is between 0
and pi/4 then A > B and f(t) > 0 for all time t. In this case the coin is always heads up and
3
we are in the range where p = 1 in theorem 2. In general, it is plausible that the probability
of heads is proportional to the amount of time which
→
N spends in the hemisphere H. This
proportion of time is precisely the p(ψ) in theorem 2 above.
M
N(t)
Tails
Heads
Figure 4: The normals to the coin lie on a circle intersecting with the equator of change of
sides.
Figures 5a and 5b show the effect of changing ψ. In Figure 5a, ψ = pi
2
and f is positive
half of the time. In Figure 5b, ψ = pi
3
and f is more often positive.
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 5a: ψ = pi/2
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
−0.5
0
0.5
1
Figure 5b: ψ = pi/3
Theorems 1 and 2 lead us to ask what is the empirical distribution of ψ when real
people toss coins. In Section 5 two empirical studies are described. The first is low-tech
and uses a coin with a thin ribbon attached. The second uses a high-speed slow motion
camera. The projection of a circle onto the plane of the camera is an ellipse. Using image
analysis techniques we fit the ellipses to the images of the tossed coin. A simple function
of the lengths of the major and minor axes gives the normal to the coin in three-space. As
4
explained, these normals spin in a circle about the angular momentum vector which stays
fixed during the coin’s flight. This gives an estimate of ψ. Two methods of estimation which
agree to reasonable approximation are given.
The empirical estimates of ψ show that naturally flipped coins precess sufficiently to
force a bias of at least .01. We find it surprising that this bias persists in the limit of
vigorously flipped coins for general densities g(ωN , t).
The structure of the rest of the paper is as follows. Section 2 reviews previous literature
and data on coin-tossing. Section 3 reviews rigid body motion and proves Theorem 1. In
section 3 we also derive an exact result for the amount of precession: the amount that the coin
turns about its normal during one revolution of the normal about the angular momentum
vector is pi cos(ψ). This is an example of a Berry phase. The limiting results of Theorem 2
are proved in Section 4. Section 5 presents our data. Section 6 presents some caveats to the
analysis along with our conclusions.
2. Previous Literature
The analysis of classical randomization devices using mechanics and a distribution on ini-
tial conditions goes back to Poincare´’s analysis of roulette [Poincare´,1896],page 122-130. This
was brilliantly continued in a sequence of studies by Hopf [Hopf,1934, Hopf,1936, Hopf,1937]
who studied Buffon’s needle, introduced various mixing conditions to prove independence of
successive outcomes and gave examples where the initial conditions do not wash out. Hopf
began a classification of low order ordinary differential equations by sensitivity to initial
conditions. While this work is little known today, [von Plato,1994] gives some further his-
tory, [Strevens, 2003] offers a philosopher’s commentary and [Engel, 1992] presents a detailed
development with extensions.
It cannot be emphasized too strongly that the results above are limiting results: Poincare´’s
arguments suggest that as a roulette ball is spun more and more vigorously the num-
bers become closer and closer to uniformly distributed. There are numerous studies (
[Barnhart, 1992], [Bass, 1985]) suggesting that real roulette may not be vigorous enough
to wash out the initial conditions.
The careful study of flipped coins was begun by [Keller, 1986], whose analysis we briefly
sketch here. He assumed that a coin flips about an axis in its plane with spin about this
axis at rate ω revolutions per second. If the initial velocity in the up direction
→
K is vz, after
t seconds, a coin flipped from initial height z0 will be at height z0 + tvz − (g/2)t2. Here
g is the acceleration due to gravity (g
.
= 32 ft/(sec)2 if height is measured in feet). If the
coin is caught when it returns to z0, the elapsed time t
∗ satisfies z0 + t∗vz − (g/2)(t∗)2 = z0
or t∗ = vz/(g/2). The coin will have revolved ωvz/(g/2) times. If this is between 2j and
2j + 1 the initial side will be up-most. If it is between 2j + 1 and 2j + 2 the opposite side
5
will be up-most. Figure 6 shows the decomposition of the phase space (ω, t) into regions
where the coin comes up as it started or opposite. The edges of the regions are along the
hyperbolae ωv/(g/2) = j. Visually, the regions get close together so small changes in the
initial conditions make for the difference between heads and tails.
35 60
0
200
400
600
800
1000
1200
1400
8010 t (10 s)
w
-2
Figure 6: Hyperbolas as defined by the various initial values of ω.
The spaces between the hyperbolas in Figure 6 have equal area.
The horizontal axis goes from t = 0.1 to t = 0.8.
[Engel, 1992] offers a way to get around asymptotic limits by deriving explicit error
terms for the approximations. Here is one of his theorems, specialized to the coin-tossing
case. Let f(ω, v) be a probability density on the ω − v plane. Thus f(ω, v) ≥ 0 and∫ ∫
f(ω, v)dωdv = 1. The marginal densities f1(ω), f2(v) are defined by
f1(ω) =
∫
f(ω, v)dv f2(v) =
∫
f(ω, v)dω.
The conditional densities fω(v) and fv(ω) are defined by
fω(v) =
f(ω, v)
f1(ω)
, fv(ω) =
f(ω, v)
f2(v)
.
Thus fω(v) is the probability density which gives the chance that a random quantity with
density f(ω, v) is in (v, v + dv) given an observed, fixed value of ω. The following theorem
applies to the case where ψ = pi/2.
6
Theorem (Engel-Kemperman). Let f(ω, v) be a probability density on the (ω, v) plane with
marginal densities f(ω), f(v), differentiable conditional densities fω(v), fv(ω). Translate f
to f(v + v0, ω + v0). Then, the probability over the heads region in Figure 6 (the Keller
coin) satisfies ∣∣∣P (heads)− 1
2
∣∣∣ ≤ 4pimin(Vv
v0
,
Vω
ω0
)
where
Vv =
∫ ∫
|f ′v(ω)|f(ω)dωdv, Vω =
∫ ∫
|f ′ω(v)|f(v)dvdω.
To see if this theorem is useful for natural coin-tosses, we carried out empirical mea-
surements similar to those reported in Section 5 below. These show that natural coin-tosses
(approximately one foot tosses of duration about 1/2 second) have initial conditions concen-
trated on
(∗) 36 ≤ ω ≤ 40, 7 ≤ vz ≤ 9
with ω measured in rev/sec and vx measured in ft/sec. Putting the uniform distribution on
the square in (∗), Engel’s Theorem gives∣∣∣P{heads} − 1
2
∣∣∣ ≤ .056.
All of the studies cited above assume the coin is caught in the hand without bouncing. An
analysis of the effect of bouncing in coin-tossing was suggested by [Vulovic and Prange, 1986].
Following Keller, they assume that the coin rotates about an axis through its plane. Thus,
the phase space is (w, t) as before. They hypothesize an explicit model for inelastic collisions
that determines the coins eventual resting place. The resulting partitioning of (ω, t) space
is surprisingly similar to Kellers (Figure 6 , above). The analysis above as displayed in
Figure 6 shows a reasonably fine partitionning of the phase space. In these regions, Vulovic
and Prange show that bouncing causes a fractal structure to appear in regions far from
zero. Then bouncing appreciably enhances randomness. [Zeng-yuan and Bin, 1985] carry
these considerations forward. They include both bouncing and air resistance. In their
model, bouncing, a very non-linear phenomena causes very pronounced sensitivity to initial
conditions. They neglect precession and so do not encounter our phenomenon of unfair coins
in the limit as (ωN , t) increase.
An intriguing analysis of coin-tossing appears in [Jaynes,1996] Section 10.3. As a physi-
cist, Jaynes clearly understands that conservation of angular momentum is the key to the
analysis of coin-tossing. With hindsight, we can find our statement following Theorem 1
of Section 1, that if the angle ψ is sufficiently acute, then the coin remains same side up
7
throughout its trajectory. This result was also described to us by Alar Toomre in a 1981
personal communication. Jaynes discusses weighted coins and coins spun on the edge. He
includes some data, in which a jar lid is tossed or spun in various ways and extreme biases
ensue.
We turn next to a different method of randomizing: coins spun on their edge. Here, the
situation is much changed. Spun coins can exhibit huge biases. The exact determination
of the bias depends in a delicate way on the shape of the coin’s edge and the exact center
of gravity. Indeed, magicians use coins with slightly shaved edges, invisible to the naked
eye, which always come up heads. While we will not pursue the details, we offer Figure
7 as evidence. This shows data provided by Jim Pitman from a class of 103 Berkeley
undergraduates who were asked to (a) toss a penny 100 times and record head or tail, (b)
spin the same penny 100 times and record head or tail. A histogram of the data appears in
Figure 7. The tossed coins are absolutely typical of fair coins, concentrated near 50 heads.
The spun coins show pronounced bias towards tails; several students had coins that came
up fewer than 10% heads. For further discussion, see [Snell et al, 2002].
spin
Fr
eq
ue
nc
y
10 20 30 40 50 60
0
5
10
15
Figure 7a: Coins spun on their edges
toss
Fr
eq
ue
nc
y
10 20 30 40 50 60
0
5
10
15
20
25
Figure 7b: Tossed Coins
If a coin is flipped up and allowed to bounce on the floor, our observations suggest that
some of the times it spins around a bit on its edge before coming to rest. If this is so, some
of the strong edge spinning bias comes into play. There may be a real sense in which tossed
coins landing on the floor are less fair than when caught in the hand. People often feel
the other way. But we suggest that this is because coins caught in the hand are easier to
8
manipulate. While this is clearly true, ruling out dishonesty, we stick to our conjecture.
Throughout, we have neglected the possibility of coins landing on their edge. We make
three remarks in this direction. First, as a youngster, the first author was involved in
settling a proposition bet where 10 coins were tossed in the air to land on the table (this
to be repeated 1000 times). On one of the trials, one of the coins spun about and landed
on its edge. [Murray and Teare, 1993] have developed an analysis of coins landing on edge.
Using a combination of theory and experiment they conclude that an American nickel will
land on its edge about one in 6000 tosses. Finally, [Mosteller, 1987] develops tools to study
the related question “how thick must a coin be to have probability 1/3 of landing on edge?”
In light of all the variations, it is natural to ask if inhomogeneity in the mass distribution
of the coin can change the outcome. [Lindley, 1981] followed by [Gelman & Nolan, 2002] give
informal arguments suggesting that inhomogeneity doesn’t matter for flipped coins caught
in the hand. Jaynes reports that 100 flips of a jar lid showed no evidence of bias. We had
coins made with lead on one side and balsa wood on the other. Again no bias showed up.
All of this changes drastically if inhomogenious coins are spun on the table (they tend to
land heavy side up). As explained above, some of this bias persists for coins flipped onto a
table or floor.
Coin-tossing is such a familiar image that it seems that someone, somewhere must have
gathered empirical data. The only extensive data we have found is [Kerrich, 1946]’s heroic
collection of 10,000 coin flips. Kerrich’s flips allowed the coin to bounce on the table so our
analysis doesn’t apply. His data does seem random (p = 1/2) for all practical purposes. Our
estimate of the bias for flipped coins is p =Ë™ .51. To estimate p near 1/2 with standard error
1/1000 requires 1
2
√
n
= 1/1000 or n
.
= 250, 000 trials. While not beyond practical reach,
especially if a national coin-toss was arranged, this makes it less surprising that the present
research has not been empirically tested.
3. Rigid Body Motion
This section sets up notation, reviews needed mechanics and proves extensions of Theo-
rems 1 and 2. Before plunging into details, it may be useful to have the following geometric
picture of a tumbling, flipping coin. Suppose the coin starts heads up with the normal
→
N
to the coin pointing upward in direction
→
K. The initial velocities determine a fixed vector→
M (the angular momentum vector). Picture this riding along with the coin, centered at the
coin’s center of gravity, staying in a fixed orientation with respect to the coordinates of the
room. The normal to the coin stays at a fixed angle ψ to
→
M and rotates around
→
M at a fixed
rate ωN . At the same time, the coin spins (or precesses) in its plane about
→
N at a fixed rate
ωpr. This description is carefully derived in Section 3.1. Theorem 1 is proved in Section 3.2
allowing the initial configuration of the coin to be in general position. The amount ∆A of
9