Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Unpredictability of Coin Tossing in the Most Sensitive Regions of the Phase Space

Discontinuity, Nonlinearity, and Complexity 4(1) (2016) 79--89 | DOI:10.5890/DNC.2016.03.006

Zengyuan Yue

Institute of Training Science and Sport Informatics, German Sport University Cologne, 50933 Cologne, Germany

Download Full Text PDF



Detailed calculations reveal that the sensitivity of the dependence of the result of coin toss (head or tail) on the initial state in the phase space is not only very inhomogeneous but also fractal. In the most sensitive regions, the number of turns of the coin is still fluctuating as the initial height changes within the atomic scale. Thus, the predictability of the toss of a real coin fails in these regions. The portion of such unpredictable regions with sub-atomic sensitivity becomes dominant in the phase space with the improvement of the elasticity of the surface. This offers one unique example in the macroscopic physics that macroscopic determinism fails due to the extreme sensitivity. This also helps to understand why the long-time accurate weather report is not possible.


  1. [1]  Poincaré, H. (1896), Calcul des probalite's, George Carre, Paris.
  2. [2]  Poincaré, H. (1914), Le Hasard, Science et Méthode, p.65, Flammarian, Paris.
  3. [3]  Prigogine, I. and Stengers, I. (1984), Order Out of Chaos , p.271, Bantam Books, USA.
  4. [4]  Hopf, E. (1934), On causality, statistics and probability, Journal of Mathematical Physics, 13, 51-102.
  5. [5]  Hopf, E. (1936), Ü ber die Bedeutung der willkürlichen Funktionen für die Wahrscheinlichkeitstheorie, Jahresbericht der Deutschen Mathematiker 46, 179-195.
  6. [6]  Hopf, E. (1937), Ein Verteilungsproblem bei dissipativen dynamischen System, Math. Annalen, Mathematische Annalen, 114, 161-186.
  7. [7]  Yue, Z. and Zhang, B. (1985), On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process, Applied Mathematics and Mechanics, 6, 193-211.
  8. [8]  Vulovic, V.Z. and Prange, R.E. (1986), Randomness of a true coin toss, Physical Review A, 33, 576-582.
  9. [9]  Keller, J.B. (1986), The probability of heads, American Mathematical Monthly, 93, 191-197.
  10. [10]  Kolota, G. (1986),What does it mean to be random? Science, 231, 1068-1070.
  11. [11]  Murray, D.B. and Teare, S.W. (1993), Probability of a tossed coin landing on edge, Physical Review E, 48, 2547-2552.
  12. [12]  Mahadevan, L. and Yong, E.H. (2011), Probability, physics, and the coin toss, Physics Today, July, 66-67.
  13. [13]  Yong, E.H. and Mahadevan, L. (2011), Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics , 79, 1195-1201.
  14. [14]  Diaconis, P., Holmes, S. and Montgomery, R. (2007), Dynamical bias in the coin toss, SIAM Review , 49, 211-235.
  15. [15]  Strzalko, J., Grabski, J., Stefanski, A. Perlikowski, P., and Kapitaniak, T. (2008), Dynamics of coin tossing is predictable, Physics Report, 469, 59-92.
  16. [16]  Mizuguchi, T. and Suwashita,M. (2006), Dymamics of coin tossing, Progress of Theoretical Physics Supplement, 161, 274-277.
  17. [17]  Yue, Z. (2014), Filaments-nets structure of the phase space of coin tossing: mechanism for sensitivity and complexity, Journal of Discontinuity, Nonlinearity and Complexity, in press.
  18. [18]  Mandelbrot, B.B. (1977), The Fractal Geometry of Nature, W. H. Freeman and Company, New York.
  19. [19]  Mandelbrot, B.B. (1977), Fractals, W. H. Freeman and Company, San Francisco.