Classical statistics mentions the significance level, alpha=0.05. It mentions beta -- (1-beta) is the power of the test to conclude the null hypothesis. Classical statistics never mentions R, the background ratio of true to false relationships in a field. While R lies in the interval [0,infinity], you could think instead about the background probability of true relationships. PLOS had an article several years ago that showed the probability a published article falsely touts a relationship as true, a probability they called the Positive Predictive Probability,
PPV = 1 / [1 + alpha / ((1 - beta) * R))]
The person designing an experiment seeks a large power, 1 - beta, so is bounded away from 0 and at most 1, so this factor becomes irrelevant (remember, the article gets published). When R is much less than alpha; eg, R=0.001 is less than 0.05, then PPV is about
R / alpha
R / 0.05
The background proportion of true relationships R dominates over alpha and over beta in the probability the relationship is true PPV.
You do a statistical test in a "field" of relationships where most of the relationships are wrong, otherwise any relationship stated has a good chance to be correct and the "field" is easy if not boring. Consider the search for some 30 genes that might cause a genetic disease out of 30,000 genes in a genome. Then R is 1 / 1000 and (about)
PPV =. 1/(1 + 0.05/(1/1000)) = 1/51 =. 0.02
That is, such published genetics articles tout relationships that are very unlikely (0.02) to be correct.
The German pharmaceutical Bayer called a large sample of published article authors, duplicated their procedures, yet found 70 percent of the publications' touted results could not be confirmed (probably wrong). Many statistical tools will give fame -- hypothesis tests or even more so data mining tools -- these are often charlatan's tools.