c ○ 2007 Society for Industrial and Applied Mathematics Dynamical Bias in the Coin Toss ∗

We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. The limiting chance of coming up this way 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.

[1]  F. e..  Calcul des Probabilités , 1889, Nature.

[2]  E. Hopf.  On Causality, Statistics and Probability , 1934 .

[3]  E. Hopf.  Ein Verteilungsproblem bei dissipativen dynamischen Systemen , 1937 .

[4]  J. E. Kerrich,et al.  An Experimental Introduction to the Theory of probability , 1946, Nature.

[5]  H. Flanders.  Differential Forms with Applications to the Physical Sciences , 1964 .

[6]  R. Leighton,et al.  The Feynman Lectures on Physics; Vol. I , 1965 .

[7]  [THE CAUSALITY]. , 1965, Medizinische Klinik.

[8]  T. Mckeown.  Mechanics , 1970, The Mathematics of Fluid Flow Through Porous Media.

[9]  F. Bookstein.  Fitting conic sections to scattered data , 1979 .

[10]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[11]  T. Foster Lindley,et al.  Is it the Coin that is Biased , 1981 .

[12]  R. Feynmann,et al.  Surely You''re Joking , 1983 .

[13]  杨骁,et al.  On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process , 1985 .

[14]  Thomas A. Bass.  The Eudaemonic Pie , 1985 .

[15]  Yue Zeng-yuan,et al.  On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process , 1985 .

[16]  Vulovic,et al.  Randomness of a true coin toss. , 1986, Physical review. A, General physics.

[17]  J. Keller.  The Probability of Heads , 1986 .

[18]  Frederick Mosteller,et al.  Fifty Challenging Problems in Probability with Solutions , 1987 .

[19]  E. Engel.  A road to randomness in physical systems , 1992 .

[20]  Murray,et al.  Probability of a tossed coin landing on edge. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  O. Faugeras.  Three-Dimensional Computer Vision , 1993 .

[22]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[23]  Leejay Wu,et al.  3D Interpretation of Conics and Or-thogonality , 1993 .

[24]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[25]  G. R. Strakosch,et al.  What goes up ... must come down. , 1994, Health facilities management.

[26]  J. Plato.  Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective , 1994 .

[27]  W. Gander,et al.  Least-squares fitting of circles and ellipses , 1994 .

[28]  C. Gilbert,et al.  Creating Modern Probability: Its Mathematics, Physics and Philosophy. , 1995 .

[29]  Simon J. Godsill,et al.  On sequential simulation-based methods for Bayesian filtering , 1998 .

[30]  L. Long,et al.  The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematicians , 1999 .

[31]  Zhengyou Zhang,et al.  Flexible camera calibration by viewing a plane from unknown orientations , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[32]  R. Blank.  What Goes Up Must Come Down? Explaining Recent Changes in Public Assistance Caseloads , 1999 .

[33]  Jun-Sik Kim,et al.  A Camera Calibration Method using Concentric Circles for Vision Applications , 2001 .

[34]  Suk Hwan Lim,et al.  A 10,000 Frames/s 0.18 μm CMOS Digital Pixel Sensor with Pixel-Level Memory , 2001 .

[35]  V. Frémont,et al.  DIRECT CAMERA CALIBRATION USING TWO CONCENTRIC CIRCLES FROM A SINGLE VIEW , 2002 .

[36]  Howon Kim,et al.  A New Camera Caibration Method using Concentric Circles for Vision Applications , 2002 .

[37]  Deborah Nolan,et al.  You Can Load a Die, But You Can't Bias a Coin , 2002 .

[38]  Feng Xiao,et al.  Experimental high speed CMOS image sensor system and applications , 2002, Proceedings of IEEE Sensors.

[39]  M. Strevens.  Bigger than Chaos: Understanding Complexity through Probability , 2003 .

[40]  Edwin Thompson Jaynes,et al.  Probability theory , 2003 .

[41]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .