Probability Tree Calculator
Compute compound event probabilities for a two-stage experiment using multiplication rule.
About Probability Tree Calculator
A probability tree represents sequential events as branches. Each path probability is the product of probabilities along its branches (multiplication rule). The total probability of any second-stage outcome is the sum across all paths reaching it (law of total probability).
Formulas
Multiplication Rule
P(A and B) = P(A) × P(B|A)
- P(A)
- First stage probability
- P(B|A)
- Second stage conditional probability
Total Probability
P(B) = P(A)×P(B|A) + P(not A)×P(B|not A)
- P(B)
- Overall probability of B
Worked example
With the default values below, Probability Tree Calculator returns P(A and B) of 0.420000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- P(Stage 1 outcome A)accepts 0–1
- 0.6
- P(Stage 2 outcome B | Stage 1 = A)accepts 0–1
- 0.7
- P(Stage 2 outcome B | Stage 1 = not A)accepts 0–1
- 0.3
Results
- P(A and B)
- 0.420000
- P(A and not B)
- 0.180000
- P(not A and B)
- 0.120000
- P(not A and not B)
- 0.280000
- P(B) total
- 0.540000
Step by step
- 1Values used
- 2Multiplication Rule
- 3Total Probability
- 4P(A and B)= 0.420000
- 5P(A and not B)= 0.180000
- 6P(not A and B)= 0.120000
- 7P(not A and not B)= 0.280000
- 8P(B) total= 0.540000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the multiplication rule?
For sequential events, P(A and B) = P(A) × P(B|A). You multiply along the branches of the tree.
How do I find the total probability of a second-stage outcome?
Sum the probabilities of all paths that end in that outcome. This is the law of total probability.