Bayes' Theorem Calculator

Update a probability with new evidence using Bayes' theorem, for example the chance a positive test is right.

Situation	One condition and a test
Prior P(A): how common it is	1 %
P(B \| A): true positive rate (sensitivity)	95 %
P(B \| not A): false positive rate	5 %

Situation

Prior P(A): how common it is

e.g. 1% of people have the condition

P(B | A): true positive rate (sensitivity)

P(B | not A): false positive rate

= 100% − specificity

P(A | B)

16.1017%

If B happens (e.g. a positive test), the chance A is true is 16.1%. Out of 10,000 people, 95 true positives vs 495 false positives.

Exact fraction. Checked exactly.

Method: Total probability, then Bayes' theorem
Worked out with: AllCalcHub exact fractions
Check: Every step recomputed with exact fractions

P(B) overall: 5.9%
P(A | not B): 0.0531%
Likelihood ratio: 19

Of the 590 positives out of 10,000

A true (true positives)95
A false (false positives)495

The probability tree

AP = 1%
- B (95%)0.95%
- not B (5%)0.05%
not AP = 99%
- B (5%)4.95%
- not B (95%)94.05%

Out of 10,000

	B (positive)	Not B (negative)
A true	95	5
A false	495	9405

About the Bayes' Theorem Calculator

Bayes' theorem combines how likely something was beforehand (the prior) with how well the evidence separates true from false cases. It explains why a 95%-accurate test for a rare condition still gives mostly false positives: when only 1% have it, false positives from the 99% outnumber true positives.

The formula

P(A|B) = P(B|A) P(A) ÷ [P(B|A) P(A) + P(B|¬A) P(¬A)]

Worked example

Prior 1%, sensitivity 95%, false-positive rate 5%

P(A|B) = 0.95 × 0.01 ÷ (0.95 × 0.01 + 0.05 × 0.99) = 16.1%

Questions people ask

Why is the answer so low?

Because the condition is rare, most positives come from the large healthy group.

What's specificity?

The true negative rate; false positive rate = 100% − specificity.