The concept of intrinsic credibility has been recently introduced to check the credibility of ‘out of the blue’ findings without any prior support. A significant result is deemed intrinsically credible if it is in conflict with a sceptical prior derived from the very same data that would make the effect just non-significant. In this paper, I propose to use Bayesian prior-predictive tail probabilities to assess intrinsic credibility. For the standard 5% significance level, this leads to a new p-value threshold that is remarkably close to the recently proposed p < 0.005 standard. I also introduce the credibility ratio, the ratio of the upper to the lower limit (or vice versa) of a confidence interval for a significant effect size. I show that the credibility ratio has to be smaller than 5.8 such that a significant finding is also intrinsically credible. Finally, a p-value for intrinsic credibility is proposed that is a simple function of the ordinary p-value and has a direct frequentist interpretation in terms of the probability of replicating an effect. An application to data from the Open Science Collaboration study on the reproducibility of psychological science suggests that intrinsic credibility of the original experiment is better suited to predict the success of a replication experiment than standard significance.