= Base-rate effect in a positive study
{title2=$\mathbb P(T\mid P)=R(1-\beta)/[\alpha+R(1-\beta)]$}
When the <prior odds> of a real effect are $R$, a study's <Type I error> is $\alpha$, and its <statistical power> is $1-\beta$, a positive report updates those odds to $R(1-\beta)/\alpha$. This follows directly from <Bayes' theorem>. A small false-positive rate does not itself make the effect's <posterior probability> high: the prior odds and power also matter. The posterior exceeds one half exactly when $R>\alpha/(1-\beta)$.
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