Hey! This is a question about induction and probability to help settle a debate!
If more thing As are observed in Group X than Group Y, and we were to take a subset of Group X and Y, is it not the case that it is more likely, by which I mean it is more probable, than Subset X has more As than Subset Y, all other things being equal?
It's POSSIBLE that subset X does not have more than subset Y , but based on what we know from the premise, is it not the case that we would say the probability of Group X having more thing As is higher? Thanks!
As it stands, your question contains some crucial ambiguities. You ask about a case where more As are observed in group X than in group Y, but it's really not clear what "observed" means here. Do you mean that quite literally, more things that are A have been, so to speak, counted in group X? And if so, were the observations random? That is: did each thing in X have an equal chance of being observed? And then there's the question of how large the subsets we take are. I assume you mean them to be equal, but you don't say and it matters a lot. If you do, mean equal size samples, are they random? That matters too. And consider this: suppose group X contains far more objects than Y. Of the 10,000 objects in X, 100 are A. Of the 20 objects in Y, 18 are A. Suppose we take a random sample of 10 from each set. Though I'm not going to work through the details, even though there are far more As in X than in Y, the random sample from Y is likely to contain more As than the same-size random sample from X. ...