Key Idea: Bayes' theorem reverses a conditional: you know P(evidence | cause) and want P(cause | evidence). It is a classic long question on Paper 1 and Paper 2, often a screening test or 'which machine' story.
Bayes' theorem
- Draw the tree — Causes first, then the evidence.
- Multiply — Multiply along each path to get the joint probabilities.
- Divide — The wanted path ÷ the total of every path ending in the evidence.
More than two causes
- Add one branch for each cause.
- The bottom of the fraction sums all the paths, e.g. three clerks.
- That total is the law of total probability.
Bayes is always one branch divided by the total for the evidence.
Clerks X, Y, Z: joint error paths 0.010, 0.012, 0.010. P(error)? 0.032.
P(clerk Z | error)? 0.010/0.032 = 5/16 = 0.3125.
What goes on the first set of branches? The causes, with their prior probabilities.
What is the denominator of Bayes' theorem? The total probability of the evidence.