When a thick cylindrical coin is tossed in the air and lands without bouncing on an inelastic substrate, it ends up on its face or its side. We account for the rigid body dynamics of spin and precession and calculate the probability distribution of heads, tails, and sides for a thick coin as a function of its dimensions and the distribution of its initial conditions. Our theory yields a simple expression for the aspect ratio of homogeneous coins with a prescribed frequency of heads or tails compared to sides, which we validate using data from the results of tossing coins of different aspect ratios.

1.
H.
Poincaré
,
Calcul des Probabilitiés
Gauthier-Villars
,
Paris
,
1912
.
2.
E.
Hopf
, “
Remarks on causality and probability
,”
J. Math. Phys.
13
,
51
102
(
1934
).
3.
E. M. R. A.
Engels
,
The Road to Randomness in Physical Systems
Springer-Verlag
,
Berlin-Heidelberg
,
1992
.
4.
J.
von Plato
,
Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective
(
Cambridge U.P.
,
Cambridge
,
1994
).
5.
J. B.
Keller
, “
The probability of heads
,”
Am. Math. Monthly
93
,
191
197
(
1986
).
6.
J. B.
Keller
, “
Dice dynamics and probability
,” Lecture notes from IMA Summer Program 1992, Minneapolis, MN.
7.
P.
Diaconis
,
S.
Holmes
, and
R.
Montgomery
, “
Dynamical bias in the coin toss
,”
SIAM Rev.
49
,
211
235
(
2007
).
8.
J.
Nagler
and
P. H.
Richter
, “
How random is dice tossing?
,”
Phys. Rev. E
78
,
036207
1
(
2008
).
9.
J.
Nagler
and
P. H.
Richter
, “
Simple model for dice loading
,”
New J. Phys.
12
,
033016
1
(
2010
).
10.
H.
Bondi
, “
The dropping of a cylinder
,”
Eur. J. Phys.
14
,
136
140
(
1993
).
11.
V.Ž.
Vulović
and
R. E.
Prange
, “
Randomness of a true coin toss
,”
Phys. Rev. A
33
,
576
582
(
1986
).
12.
J.
Strazałko
,
J.
Grabski
,
P.
Perlikowski
,
A.
Stefański
, and
T.
Kapitaniak
,
Dynamics of Gambling: Origins of Randomness in Mechanical Systems
(
Springer-Verlag
,
Berlin Heidelberg
,
2009
).
13.
D. B.
Murray
and
S. W.
Teare
, “
Probability of a tossed coin landing on edge
,”
Phys. Rev. E
48
,
2547
2552
(
1993
).
14.
F.
Mosteller
,
Fifty Challenging Problems in Probability With Solutions
(
Dover
,
New York
,
1987
).
15.
J.
Bertrand
,
Calcul des Probabilitiés
(
Gauthier-Villars
,
Paris
,
1889
).
16.
E. T.
Jaynes
, “
The well-posed problem
,”
Found. Phys.
3
,
477
493
(
1973
).
17.
L.
Landau
and
E.
Lifschitz
,
Mechanics
, 3rd ed. (
Pergamon
,
Oxford, UK
,
1976
).
18.
T.
Stoppard
,
Rozencrantz and Guildenstern are Dead
(
Grove
,
New York
,
1968
).
19.
L.
Mahadevan
and
E. H.
Yong
, “
Probability, physics, and the coin toss
,”
Phys. Today
64
(
7
),
66
67
(
2011
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.