Bayes' Theorem
What is Bayes’ Theorem
- Bayes’ Theorem allows you switch the order of conditional probabilities
- If you know , and then Bayes’ Theorem allows you to find
- 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:
- 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: and
Exam Tip
- 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.
b)
Given that Lucy answered a question correctly, find the probability that it was about TV.