Exercise · ZW-BAYES-THEOREM-010 · ★★★ Advanced

A disease affects 1 in 1000. The test is positive for 99 % of the sick and 1 % of the healthy. If you test positive, what is roughly the probability you are sick?

Choose one option.

Show answer

A. ≈ 0.09

Explanation

\(\dfrac{0.001\cdot0.99}{0.001\cdot0.99+0.999\cdot0.01}=\dfrac{11}{122}\approx0.09\): false positives outnumber true ones about ten to one.

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Review the concept: Bayes' theorem

Bayes' theorem inverts a conditional probability: it gives the probability \(P(B_j\mid A)\) of a cause \(B_j\) given an observed effect \(A\), from the probability of the effect given that cause.

\[P(B_j\mid A)=\frac{P(A\mid B_j)\,P(B_j)}{\sum_{i}P(A\mid B_i)\,P(B_i)}\]

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