Visual Bayes' Theorem & Diagnostic Decision Tree Solver
Bayes' Theorem updates prior belief $P(A)$ given evidence $B$ via $P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B \mid A)P(A) + P(B \mid \neg A)P(\neg A)}$, dividing the True Positive branch by the sum of all positive paths.
P(A | +) = [P(+ | A)·P(A)] / [P(+ | A)·P(A) + P(+ | ¬A)·P(¬A)]Inputs & Parameters (3)
20-Row Comparison Table
| # | Prior Base Rate P(A) (%) | Posterior PPV P(A | +) | Total P(+) | False Pos Share |
|---|---|---|---|---|
| 01 | 0.5 % | 5.631 % | 8.44% | 94.4% |
| 02 | 5.5 % | 40.868 % | 12.78% | 59.1% |
| 03 | 11 % | 59.476 % | 17.57% | 40.5% |
| 04 | 16 % | 69.343 % | 21.92% | 30.7% |
| 05 | 21 % | 75.942 % | 26.27% | 24.1% |
| 06 | 26.5 % | 81.066 % | 31.05% | 18.9% |
| 07 | 31.5 % | 84.522 % | 35.41% | 15.5% |
| 08 | 37 % | 87.46 % | 40.19% | 12.5% |
| 09 | 42 % | 89.582 % | 44.54% | 10.4% |
| 10 | 47 % | 91.327 % | 48.89% | 8.7% |
| 11 | 52.5 % | 92.92 % | 53.67% | 7.1% |
| 12 | 57.5 % | 94.14 % | 58.02% | 5.9% |
| 13 | 62.5 % | 95.19 % | 62.38% | 4.8% |
| 14 | 68 % | 96.188 % | 67.16% | 3.8% |
| 15 | 73 % | 96.979 % | 71.51% | 3% |
| 16 | 78.5 % | 97.746 % | 76.3% | 2.3% |
| 17 | 83.5 % | 98.363 % | 80.64% | 1.6% |
| 18 | 88.5 % | 98.918 % | 85% | 1.1% |
| 19 | 94 % | 99.465 % | 89.78% | 0.5% |
| 20 | 99 % | 99.915 % | 94.13% | 0.1% |
Key Analytical Takeaways
- 01Even with a $95\%$ sensitive test and only an $8\%$ false-positive rate, a $5\%$ base rate yields a Posterior Positive Predictive Value $P(A \mid +)$ of just $38.5\%$!
- 02This happens because the $8\%$ false-positive rate acts on the huge $95\%$ healthy population ($7.6\%$ of total), outweighing true positives ($4.75\%$ of total).
- 03The 4 terminal leaves of the probability tree always sum to exactly $100.0\%$ of the population.
Frequently Asked Questions
What is the Base-Rate Fallacy in medical testing and spam filtering?+
People intuitively confuse P(+ | Disease) with P(Disease | +). When a condition is rare (low prior P(A)), false positives from the large negative population can easily outnumber true positives.
How is Specificity related to the False Positive Rate?+
Specificity (True Negative Rate, P(− | ¬A)) is the exact complement of the False Positive Rate: Specificity = 100% − False_Positive_Rate.
What is the base-rate fallacy in diagnostic testing?+
Even with 99% sensitivity and 99% specificity, testing a population where prevalence P(A) is 0.1% produces roughly 10 false positives for every 1 true positive, leaving positive predictive value P(A|+) below 10%.