Determine PPV from type I and type II error rates

Consider these two scenarios:

In both of these scenarios, flipping heads 60 times (the X-axis should read # of heads, not # of coin flips) would be reported as a p of ~0.05 if we are using the fair coin as a null hypothesis for our one-sample proportion test. However, the likelihood of that p value actually representing a true finding is much higher for the second scenario compared to the first. The difference, of course, is the power of the study (the power being much higher for the totally rigged coin study because the effect size difference is much larger).

I want to be able to quantify exactly how much greater the likelihood of a true finding is in the second scenario compared to the first. This would be the positive predictive value (PPV). So essentially, I'm asking how can I relate a known $$\alpha$$ of 0.05 and a calculated $$1-\beta$$ to determine a PPV?