Base-rate effect in a positive study ($\mathbb{P}(T\mid P)=R(1-\beta )/[\alpha +R(1-\beta )]$)

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 $R>\alpha /(1-\beta )$.