Probability Tree Builder

Build a two-stage probability tree, see every branch multiply out to a joint probability, and reverse the conditioning with Bayes' theorem.

Your tree

First event (A)

Second event (B)

P(Has condition)

P(Tests positive | Has condition)

P(Tests positive | not Has condition)

Enter probabilities as decimals between 0 and 1, for example 0.25 for 25%.

Probability tree

P(Tests positive)

0.27

P(Has condition | Tests positive)

0.33333

P(Has condition | not Tests positive)

0.0137

P(Tests positive) = 0.09 + 0.18 = 0.27
P(Has condition | Tests positive) = 0.09 ÷ 0.27 = 0.33333

Out of 1,000 cases: about 90 are "Has condition and Tests positive", 10 are "Has condition and not Tests positive", 180 are "not Has condition and Tests positive", and 720 are "not Has condition and not Tests positive". Of the 270 with Tests positive, 90 also have Has condition, which is 33.33%.

How to use this tool

Pick a preset, or name your two events and enter the three probabilities: P(A), P(B given A), and P(B given not A).
The tree shows each branch probability and, at the end of each path, the joint probability found by multiplying along the path.
Read the total probability of B and the reversed probabilities such as P(A given B).
Use the "out of 1,000" sentence to see the same numbers as counts.

Formula Reference

Joint probability

P(A and B) = P(A) × P(B | A)

Total probability

P(B) = P(A)P(B|A) + P(not A)P(B|not A)

Bayes' theorem

P(A | B) = P(A and B) ÷ P(B)

ArticleProbability Trees ArticleBayes' Theorem ArticleJoint, Marginal & Conditional Probability ArticleSensitivity, Specificity & ROC Curves CalcBayes' Theorem Calculator GlossaryGlossary: Probability Tree

Frequently Asked Questions

Why do I multiply along the branches?+

Each branch after the first is a conditional probability. The chance of following a whole path is the chance of the first branch times the conditional chance of the second, which is the multiplication rule P(A and B) = P(A) × P(B | A).

Why do the probabilities on the branches from one node add to 1?+

Because from any node exactly one of the branches must happen. The branch for B and the branch for not B together cover every possibility.

What do the preset numbers show?+

The medical-test preset has a 10% prevalence, 90% sensitivity and a 20% false-positive rate. Even so, only one in three positive results is a true case, because there are so many more healthy people than sick ones. This is the base-rate effect.

How is this different from the Bayes' theorem calculator?+

The tree shows all four joint outcomes and how they combine. The Bayes calculator gives just the reversed probability. Both use the same arithmetic.

How to use this tool

Pick a preset, or name your two events and enter the three probabilities: P(A), P(B given A), and P(B given not A).

The tree shows each branch probability and, at the end of each path, the joint probability found by multiplying along the path.

Read the total probability of B and the reversed probabilities such as P(A given B).

Use the "out of 1,000" sentence to see the same numbers as counts.