Law of total probability & tree diagrams
The law of total probability splits an event across a partition: P(B) = Σ P(B|Aᵢ)P(Aᵢ). A tree diagram is the same computation drawn out — branch probabilities are conditional, you multiply along a path and add across paths. This is the standard setup for two-stage experiments and the denominator of every Bayes calculation.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Choose a partitionMutually exclusive events covering everything: which urn, which machine, which route. This is the first branching of the tree.
- Label branches with conditional probabilitiesSecond-stage branches are conditional on the first, so they need not sum to 1 across the whole diagram — only within each fan.
- Multiply along a pathA complete path gives P(A ∩ B) = P(A)P(B|A).
- Add across pathsSum every path leading to the event of interest. That sum is the law of total probability.
Worked example
Urn A has 3 red and 2 blue balls; urn B has 1 red and 4 blue. An urn is chosen at random and a ball drawn. Find P(red).
- P(A) = P(B) = 1/2.
- P(red|A) = 3/5 and P(red|B) = 1/5.
- Path via A: (1/2)(3/5) = 3/10. Path via B: (1/2)(1/5) = 1/10.
- Add: 3/10 + 1/10.
Answer. P(red) = 4/10 = 2/5.
Where marks get dropped
These are the specific errors that cost credit on law of total probability & tree diagrams questions — QED's rubric penalises each of them separately.
- Using unconditional probabilities on the second-stage branches. Those branches are conditional on the first stage.
- Choosing a partition that overlaps or does not cover everything. The law requires a genuine partition.
- Adding along a path instead of multiplying. Multiply down a path; add across paths.
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Law of total probability & tree diagrams — frequently asked questions
How does this connect to Bayes?
The law of total probability produces the denominator P(B), and Bayes divides one path by that total. Working backwards up the tree IS Bayes.
Must the branch probabilities sum to 1?
Within each fan of branches from one node, yes. Across the whole diagram, no — the path probabilities are what sum to 1.
Can I use it with more than two stages?
Yes — the tree just gets deeper, and each complete path multiplies all its conditional probabilities together.
The rest of Probability
Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Sample spaces & events
- 2Conditional probability & independence
- 3Bayes’ theorem
- 4Random variables & expected value
- 5Variance & standard deviation
- 6Binomial & uniform distributions
- 7Law of total probability & tree diagrams
- 8Geometric & Poisson distributions
- 9Joint, marginal & conditional distributions
- 10Markov & Chebyshev bounds
- 11Equally likely outcomes & counting
- 12Linearity of expectation
- 13Indicator random variables
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