I agree on your point, if we are sufficiently creative we can get many extreme results. For example, I made an addition to the first paragraph of my previous comment, that you might have missed, giving an example where the probability of the null hypothesis being true when p=0.05 is zero (or arbitrarily small, if we replace the discrete probability lump under the alternative hypothesis by a continuous distribution which is concentrated enough). I also added a comment on the second paragraph, by the way.<p>One minor comment on your example. If H0:fair coin and H1:two-headed and the statistic is the number of heads h, I cannot reject (at the 0.05 level) the null when n (the number of flips) is small even if I&#x27;m only getting heads. For one flip, p[h=1|H0]=0.5. For two flips p[h=2|H0]=0.25. For n&gt;5 you will of course reject the null hypothesis for every case where H1 is true (and for ~5% of the cases where H0 is true). There will be no false negatives. But I guess you have noticed that this doesn&#x27;t help with the false discovery rate in this example: when H1 is true the p-value will be very small (1&#x2F;2^n) so if the observed p-value is ~0.05 (or any other value larger than 1&#x2F;2^n) then it&#x27;s for sure a false positive (because there will be at least one occurrence of tails).<p>Ok, enough time wasted on this subject :-)