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: Bayes' theorem

      Rachel Thomas
      25 January, 2016
      1 comments

      Suppose that a particular type of cancer affects 1% of the population. There is a test for this cancer but it's not perfect: although the test gives a positive result for 90% of people who have the cancer, it also gives a positive result for 5% of the people who are cancer-free. You have just received a positive test result – what is the probability you have cancer?

      Many of us would say there is now a 90% chance that we have cancer. But this isn't correct – your chances are closer to 15%. To understand why we have to call on conditional probabilities and a very useful result: Bayes' theorem.

      A conditional probability is the probability that one thing is true (in this example, that you have this type of cancer) given another thing is true (your test result is positive). For our example we'd write the conditional probability of having this cancer given a positive test result as P(cancer|positive).

      Before you had the test, you believed that your probability of having this cancer was P(cancer)=0.01. So, in a population of 10,000 people you'd expect 100 of them to have this cancer. This group of people is represented by the red circle in the picture.

      Now you've had a positive test result. How many people out of our population of 10,000 will have had a positive test result – represented by the blue circle in the picture?

      There is a 90% chance of a positive test result if you have cancer. For our example population of 10,000 people, 90 out of 100 people with this cancer will receive a positive test result – these people lie in the intersection of the blue and red circles.

      And there is a 5% chance that you'll still get a positive test if you are cancer-free – these people lie in the blue circle that is outside of the red circle in the picture. So for the 9,900 cancer-free patients in our population, 495 will incorrectly test positive. This gives a total of 90+495=585 people out of every 10,000 people expected to get a positive test.

      So what is P(cancer|positive), the probability of you having this cancer, given you've had a positive test result? This is the proportion of people who have cancer and have had a positive test result (the intersection of the two circles) of all the people who've had a positive test result (the blue circle): 90/585=0.154. Or written in terms of probabilities P(cancer|positive)=P(cancer)P(positive|cancer)P(positive)=0.01×0.90.0585=0.154. where P(positive|cancer) is the probability of getting a positive test result given you do have cancer.

      So your chance of having this cancer given you've had a positive test result is a much more encouraging 15%. This result is known as Bayes' Theorem, written more generally as P(A|B)=P(A)P(B|A)P(B) Bayes' theorem allows you to update your prior belief (in this case, that your chance of having cancer was 1%) when new evidence becomes available (a positive test result).

      You can read more about conditional probability and Bayes' theorem on Plus.


      This article now forms part of our coverage of the cutting-edge research done at the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge. The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more.

      INI logo

      • Log in or register to post comments

      Comments

      math.nights

      7 March 2016

      Permalink

      To Arabic: https://goo.gl/3PZ0fU

      • Log in or register to post comments

      Read more about...

      bayes theorem
      conditional probability
      medical statistics
      Maths in a minute
      INI
      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