Probability tree diagram example: a screening test

What it shows

A two-stage probability tree for a breast cancer screening test, using the mammography example of Gigerenzer and colleagues (2007). The first stage splits screened women by condition; the second splits each branch by test result.

Each branch carries a conditional probability, and each end of the tree carries the joint probability of its path and the cell of the 2 × 2 table it belongs to. A panel below the tree works out the positive predictive value.

How to read it

1. 1

   Start at the root. With a prevalence of 1%, the first split is 0.01 (cancer) and 0.99 (no cancer).

2. 2

   Each second-stage branch is conditional on the first: given cancer, positive 0.90 and negative 0.10; given no cancer, positive 0.09 and negative 0.91.

3. 3

   Multiply along a path for its joint probability. True positives are 0.01 × 0.90 = 0.0090; false positives are 0.99 × 0.09 = 0.0891.

4. 4

   The positive predictive value is true positives over all positives: 0.0090 / 0.0981 = 0.092, about 9%.

The numbers behind it

• Prevalence 1%; sensitivity 90%; false-positive rate 9% (specificity 91%).
• Joint probabilities: true positive 0.0090, false negative 0.0010, false positive 0.0891, true negative 0.9009; they sum to 1.0000.
• Positive predictive value: 0.0090 / (0.0090 + 0.0891) = 0.092.

Ways to use it in teaching

• Ask students for their estimate of the chance that a woman with a positive result has cancer, before showing the panel, and compare it with the 9%.

• Rebuild the tree with a higher prevalence, as in a high-risk clinic, to show how the predictive value depends on the base rate.

• Use it to teach the base-rate fallacy in an evidence-based practice or research ethics session.

Sources

• Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8(2), 53–96. https://doi.org/10.1111/j.1539-6053.2008.00033.x
• Positive and negative predictive values. (n.d.). In Wikipedia. Retrieved October 19, 2026, from https://en.wikipedia.org/wiki/Positive_and_negative_predictive_values
• Tree diagram (probability theory). (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Tree_diagram_(probability_theory)