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
    • Maths in a minute: The binomial distribution

      Christine Currie
      13 July, 2017
      Here's a bet. I flip a coin a total number n of times and you win £1000 if it comes up heads k times, otherwise you lose £1000. What's the probability of you winning for different values of n and k?
      Plane

      The binomial distribution for different pairs of values of n and k. The graph plots the probability that k out of n trials are a success against the value of k.

      The answer is given by the binomial distribution. Phrased in more general terms, the distribution gives the probability of achieving k successes (eg getting heads) in a total number of n trials of a process (eg flipping a coin). To work out the distribution we need to know the probability of a success. In our coin example, this would be 0.5 for a fair coin, but could be another value for a crooked coin. To capture the general case let's just write p for that probability. The probability of observing a particular sequence of successes and failures in a particular order is equal to the product of the individual probabilities. For example, the probability of observing the sequence Success, Success, Failure is p×p×(1−p).

      But we don't mind what order our successes and failures come in, we just care about the total number of successes. That means we need to add up the probabilities of each sequence that results in the right number of successes. Going back to the example of three trials and two successes, we could have

      • Success, Success, Failure
      • Success, Failure, Success
      • Failure, Success, Success.
      Each of these has a probability p×p×(1−p), so the probability of one of the three possible outcomes occurring is 3×p×p×(1−p)=3p2(1−p). Writing this more generally, P(k successes from n trials)=(nk)pk(1−p)n−k. Here the variable k can take values 0, 1, 2, up to n, and the first term is the binomial coefficient (nk)=n!(n−k)!k!. The exclamation mark denotes the factorial which is defined as n!=n×(n−1)×(n−2)...×2×1.

      The binomial coefficient can be thought of as the number of ways of choosing an unordered list of k outcomes from n possibilities. You can find out more here.

      The binomial distribution has mean np and variance np(1−p). To see a worked example of the binomial distribution in action, read this article.
      • Log in or register to post comments

      Read more about...

      Maths in a minute
      probability
      probability distribution
      binomial distribution
      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