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Bayes and conditional probability

Conditional probability is the probability of $A$ when you already know that $B$ has happened. Bayes' theorem reverses the condition: from the probability of a positive test given the disease, to the probability of the disease given a positive test. You first need the total probability of a positive test. If the disease is rare, most positive tests can still belong to healthy people.

P(A\mid B) = \frac{P(A \cap B)}{P(B)}

conditional probability

P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)}

Bayes

Symbols

$P(A)$	the probability of A
$P(Bmid A)$	the probability of B given A

Example

Two percent are sick. The test catches 90% of the sick and 5% of the healthy. $P(\text{positive}) = 0.02\cdot 0.9 + 0.98\cdot 0.05 = 0.067$ . Bayes gives $\dfrac{0.018}{0.067} \approx 0.2687$ .

The denominator is everyone who tests positive, both sick and healthy.

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Part of Mathematics R1: Probability and combinatorics.