Bayes’ Theorem
Update beliefs with evidence.
Key idea
Update beliefs with evidence.
In depth
P(A | B) = P(B | A)P(A)/P(B). It updates a prior belief using evidence. P(B) is often found with the law of total probability.
Worked example
Using the factory above, P(machine Y | defect) = 0.02/0.032 = 0.625.
Formula
Real-world uses
- Updating spam filter rules with new evidence
- Interpreting medical screening results
Common mistake
Confusing P(A | B) with P(B | A), as in the prosecutor’s fallacy.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Bayes’ Theorem?
Update beliefs with evidence. It is a topic at the expert level in Probability & Statistics.
What is the formula for Bayes’ Theorem?
The key formula is P(A|B) = P(B|A)P(A)/P(B).
Where is Bayes’ Theorem used in real life?
Real-world examples: Updating spam filter rules with new evidence; Interpreting medical screening results.
What should I learn before Bayes’ Theorem?
Make sure you are comfortable with Law of Total Probability, the previous topic in Probability & Statistics.
What is a common mistake with Bayes’ Theorem?
Confusing P(A | B) with P(B | A), as in the prosecutor’s fallacy.
How can I practise Bayes’ Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Bayes’ Theorem?
Move on to Medical Tests and Base Rates next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Medical Tests and Base Rates – Why positive results can still be probably false.
- Linearity of Expectation – Expectation of sums is sum of expectations.