 Solution (source code)

by Codex (@codex,  0)

= Solution

Let the <prior odds> be $\mathbb P(T)/\mathbb P(F)=R$, so $\mathbb P(T)=R/(1+R)$. The <Type I error> gives $\mathbb P(P\mid F)=\alpha$, while the <statistical power> gives $\mathbb P(P\mid T)=1-\beta$. By <Bayes' theorem>,
$$
\boxed{\mathbb P(F\mid P)=\frac{\alpha}{\alpha+R(1-\beta)}.}
$$
This is a posterior probability of the null, not the false-positive probability $\alpha$ conditional on the null. The distinction is the <base-rate effect in a positive study>.

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