Bayes' theorem calculator

How to use it

1. Enter P(A), the probability of the hypothesis before the new evidence.
2. Enter P(B|A), how likely the evidence is when A is true.
3. Pick what else you know: P(B|not A) or P(B), and fill that box.
4. Read P(A|B), the updated probability.

How it works

Bayes’ theorem is P(A|B) = P(B|A) × P(A) / P(B). When you give P(B|not A) instead of P(B), the calculator first finds P(B) = P(B|A) × P(A) + P(B|not A) × (1 − P(A)). The “change from the prior” row is P(A|B) divided by P(A).

Worked example

A condition affects 1% of people. A test is positive for 90% of those who have it and for 5% of those who do not.

• P(B) = 0.9 × 0.01 + 0.05 × 0.99 = 0.0585
• P(A|B) = 0.009 / 0.0585 ≈ 0.1538, or 15.38%

So only about 15% of positive results are true positives.

Tips

• Keep “A” and “B” straight: P(A|B) and P(B|A) are different numbers.
• If P(A) is 0 or 1, the evidence cannot change it.