Direct answer
A probability tree multiplies conditional probabilities along a path to obtain a joint outcome and adds disjoint path probabilities when several paths answer the same final question.
Visual explanation
Multiply paths and add disjoint leaves
What this calculation tells you
A tree makes sequential conditioning explicit. Each node is a current sample space, each branch is conditional on the path already taken, and each leaf represents one disjoint complete outcome.
Name the sequence and define every node before assigning probabilities. If order is irrelevant or the process does not unfold conditionally, a table or set diagram may communicate the model more clearly.
Where it is used
Diagnostics and screening
Display arithmetic for entered conditional rates without making a diagnosis.
Operations
Model staged routing, inspection, or conversion paths.
Reliability
Represent explicit event sequences and conditional branches.
Education
Link conditional, joint, total, and posterior probability visually.
When this guide helps
- Modeling two-stage outcomes.
- Calculating total probability across paths.
- Auditing whether branches total one.
- Explaining where a posterior denominator comes from.
Start with the statistical question
Name the sequence and define every node before assigning probabilities. If order is irrelevant or the process does not unfold conditionally, a table or set diagram may communicate the model more clearly.
Branches grow from a root through two stages. Selecting a leaf highlights the multiplication path; selecting a final event highlights several leaves whose probabilities are added.
Worked example
If 40% enter path A and 20% of A outcomes succeed, that leaf has probability 0.4×0.2=0.08. If 60% enter not-A and 50% of those succeed, total success is 0.08+0.30=0.38.
Assumptions that carry the result
Every branching set must be mutually exclusive and exhaustive at its node. Conditional probabilities must refer to the exact path reaching that node and cannot be borrowed from an unmatched population.
Interpret the result without overreaching
A tree can display an incorrect model beautifully. It does not validate independence, stationarity, causal order, or the quality of entered probabilities.
- Using unconditional probabilities on later branches.
- Forgetting a branch needed to complete probability mass.
- Adding paths that overlap rather than representing disjoint leaves.
Worked case: two production lines
Line A makes 60% with 2% defects; line B makes 40% with 5% defects.
Defect paths are 0.60 × 0.02 = 0.012 and 0.40 × 0.05 = 0.020; total = 0.032.
Modeled defect probability is 3.2%.
Branch rates are assumptions or estimates, not guarantees.
Reproduce this worked caseOpen Probability Tree Calculator
Worked case: identify source given defect
Use the same tree.
P(B and defect) = 0.020; divide by total defect 0.032 to get 62.5%.
Given a modeled defect, probability it came from B is 62.5%.
This reverse probability needs the base production shares.
Reproduce this worked caseOpen Probability Tree Calculator
probability trees: compare assumptions, not just answers
Every set of branches from one node must reconcile to one. Keep conditional labels and time/population basis explicit.
| Case | Calculation focus | Interpretation |
|---|---|---|
| Total defect | Sum two paths | 3.2% |
| B | defect | 0.020/0.032 | 62.5% |
probability trees: calculation checklist
- Branches exhaustive/disjoint
- Sibling probabilities sum 1
- Multiply along paths
- Add disjoint paths
- Base rates retained
Practical questions
Frequently asked questions
Do I multiply or add in a tree?
Multiply probabilities along one complete path; add probabilities of disjoint complete paths that all satisfy the requested event.
Must every tree be chronological?
No, but each level must represent a coherent sequence of conditioning information.
How do I know branches are complete?
Outgoing branches from each node should describe mutually exclusive outcomes whose conditional probabilities total one.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.