 This answer is indeterminate if we don’t take into account the base-rate of truth, viz., the incidence rate of truth in the population of theories. If the base rate is very low (let’s assume that only 1 in 50 theories have been true), then it is unlikely that T is true given success. \(\textrm{Prob}(T\mid S)\) would be around 0.3. But this does not imply something about the connection between success and truth. It is still the case that the false positives are low and that the true positives high. The low probability is due to the fact that truth is rare (or that falsity is much more frequent). For \(\textrm{Prob}(T\mid S)\) to be high, it must be the case that \(\textrm{Prob}(T)\) is not too small. But if \(\textrm{Prob}(T)\) is low, it can dominate over a high likelihood of true positives and lead to a very low posterior probability \(\textrm{Prob}(T\mid S)\). Similarly, the probability that a theory is false given that it is successful (i.e., \(\textrm{Prob}({\textrm{not-}T}\mid S))\) may be high simply because there are a lot more false theories than true ones. As Peter Lewis put it: