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Bayes' theorem

Bayes' theorem flips a conditional probability: from $P(B|A)$ you find $P(A|B)$ . It is widely used in testing, where you know the probability of a positive result given disease, but want the probability of disease given a positive result.

P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}

Bayes' theorem

Symbols

$P(A)$	probability of $A$ before knowing about $B$
$P(A\|B)$	probability of $A$ given that $B$ has occurred

Example

6% defect rate, the test finds the defect 99% of the time and false-positives 10% of the time on good units.

$P(\text{defect}|\text{positive}) = \dfrac{0.99\cdot 0.06}{0.99\cdot 0.06 + 0.1\cdot 0.94} \approx 0.387$ .

Always compute the denominator

P(B)

by summing over all the ways

B

can happen (positive test, sick or healthy).

Practise probability for free →

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Part of Statistics and Risk Analysis: Probability.