Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Solution to the coin tossing problem

      1 January, 1998
      January 1998

      The chance that any fixed set of k+1 tosses gives k heads followed by a tail is:

      (1/2)^(k+1)

      In n tosses there are n-k possible starting points of such sequences. Therefore the expected number of wins is:

      (n-k)*(1/2)^(k+1)

      Using probability theory, one finds that the actual number of wins has what is called an approximate Poisson distribution. That is to say, the probability that you win exactly r pennies is given approximately by the Poisson probability:

      p[r] = lambda^r/r! * e^(-lambda), lambda = (n-k)*(1/2)^(k+1)

      Your average number of wins equals lambda. If lambda is big, then the number of wins will generally be big also, and if lambda is small then there will be only a small number of wins (perhaps 0).

      The threshold between big and small occurs when lambda is a 'reasonable' number, say the value 1. When lambda = 1, then k is very close to log2(n). In this threshold case the Poisson probabilities are given in the following table.

      p[0]=0.368, p[1]=0.368, p[2]=0.184, p[3]=0.061, p[4]=0.015, p[5]=0.003

      Substantially longer runs than log2(n) are exceedingly unlikely, while substantially shorter runs are commonplace.

      Acknowledgements

      This solution was written by Geoffrey Grimmett. You may also be interested in his article "What a coincidence!" elsewhere in this issue.

      • Log in or register to post comments
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms