Suppose we sample a person from the population. They are a member of US Congress. We define the null hypothesis $H_0$ as "the person is American". We calculate the $p$-value: $P[member\ of\ Congress | American] \ll 0.05$. Since if the null hypothesis holds, the person is very unlikely to be a member of Congress, we reject the null hypothesis and decide that the person is very likely not an American. This conclusion is obviously very wrong, as all members of Congress are American.
Which assumptions of the hypothesis testing did I violate here? In other words, if I encounter a similar (but more obscure) application where this methodology is also not appropriate, how do I identify it?
Edit: to clarify the methodology. In Fisher's example, where he is trying to identify if the tea-drinking lady can correctly identify cups of tea, he considers the chances of her guessing all cups of tea correctly if she had no ability:
Given that if she had no ability, it would be very unlikely for her to guess all cups of tea correctly, he rejects the hypothesis of her having no ability. So the conclusion is that it is unlikely that she has no ability and just got lucky. Which makes perfect sense.
In my example, given that if the person was an American, it would be very unlikely for them to be a member of Congress, we reject the hypothesis of them being an American. So we conclude that it is unlikely that they are an American and we just got lucky picking a Congressman out of all Americans.
So is the reason for this paradox the formulation of the alternative hypothesis? To reach the paradox, we have to assume that the alternative hypothesis forms a disjoint set with the null hypothesis. E.g. alternative of "no ability" is "has ability"; alternative of "population of Americans" is "population of non-Americans". Whereas all we can say is that the alternative hypothesis is some different population than the null hypothesis - it could be a subset! E.g. alternative to "No ability" could be "Someone told her the correct ordering", and alternative to "population of Americans" could be "population of Americans in Washington, DC"? So rejecting the null hypothesis is correct in both cases - it is indeed unlikely that we sampled from all Americans and stumbled on a US congressman - and the paradox arises from jumping to a particular alternative hypothesis and "accepting" it for no good reason? While in fact we cannot say anything about any alternative from the experiment - ALL we know is that the null was unlikely to cause the observation!