Bayes rule with a base rate

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Problem

A condition has prevalence $P(D)=0.10$ . A test has sensitivity $P(+\mid D)=0.80$ and false-positive rate $P(+\mid D^c)=0.20$ . Find $P(+)$ and $P(D\mid +)$ . Are the events $D$ and $+$ independent?

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Total probability gives

P(+)=0.8(0.1)+0.2(0.9)=0.26=\frac{13}{50}

Bayes rule gives

P(D\mid +)=\frac{P(+\mid D)P(D)}{P(+)} =\frac{0.08}{0.26}=\frac4{13}

They are not independent because $P(+\mid D)=0.80\ne P(+)=0.26$ .

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Bayes rule with a base rate

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