Bayes' Theorem (DP IB Analysis & Approaches (AA)) : Revision Note

Dan Finlay

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Bayes' Theorem

What is Bayes’ Theorem

  • Bayes’ Theorem allows you switch the order of conditional probabilities

    • If you know straight P left parenthesis B right parenthesisstraight P left parenthesis B apostrophe right parenthesis and straight P left parenthesis A vertical line B right parenthesis then Bayes’ Theorem allows you to find straight P left parenthesis B vertical line A right parenthesis

  • Essentially if you have a tree diagram you will already know the conditional probabilities of the second branches

    • Bayes’ Theorem allows you to find the conditional probabilities if you switch the order of the events

  • For any two events A and B Bayes’ Theorem states:



    • This is given in the formula booklet

    • This formula is derived using the formulae:

      •  

      •  straight P invisible function application open parentheses A close parentheses equals straight P invisible function application open parentheses B intersection A close parentheses plus straight P invisible function application open parentheses B apostrophe intersection A close parentheses

      • and 

  • Bayes’ Theorem can be extended to mutually exclusive events B1, B2, ..., Bn and any other event A

    • In your exam you will have a maximum of three mutually exclusive events



      • This is given in the formula booklet

How do I calculate conditional probabilities using Bayes’ Theorem?

  • Start by drawing a tree diagram

    • Label B1 & B2 (& B3 if necessary) on the first set of branches

    • Label A & A’ on the second set of branches

  • The questions will give you enough information to label the probabilities on this tree

  • Identify the probabilities needed to use Bayes’ Theorem

    • The probabilities will come in pairs: straight P open parentheses B subscript i close parentheses and straight P open parentheses A vertical line B subscript i close parentheses

4-3-3-ib-aa-hl-bayes-theorem

Examiner Tips and Tricks

  • In an exam you are less likely to make a mistake when using the formula if you draw a tree diagram first

Worked Example

Lucy is doing a quiz. For each question there’s a 45% chance that it is about music, 30% chance that it is about TV and 25% chance that it is about literature. The probability that Lucy answers a question correctly is 0.6 for music, 0.95 for TV and 0.4 for literature.

a) Draw a tree diagram to represent this information.

4-3-3-ib-aa-hl-bayes-theorem-a-we-solution

b) Given that Lucy answered a question correctly, find the probability that it was about TV.

4-3-3-ib-aa-hl-bayes-theorem-b-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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