Non sequitur of the month: personal communication

I'll describe this month's entry in abstract terms first, then provide two examples. Suppose you have two sets A and B. They're both non-empty and they overlap; other than that, no additional assumptions about them are made. Our non sequitur of the month is the following implication:

If the number of elements that belong to the intersection of A and B constitutes x percent of the number of all elements of A, then it also constitutes x percent of the number of all elements of B.

This is true if and only if A and B are exactly equal in size (or, as a special case, if they are one and the same set). But no such assumption was made, so it is definitely not true; in fact, it is a stunning non sequitur.

I've encountered this baffling mistake in two real-life contexts; describing those may help highlight how incredibly wrong this reasoning is. First context: There are studies trying to estimate the percentage of homosexual men among pedophiles. Any study that comes up with such an estimate (say 10 percent) will invariably be quoted somewhere in the media as saying that 10 percent of homosexual men are pedophiles--as if it were the same thing. Second context is a guy I knew named Bob (not his actual name). Bob is a very successful individual. He holds a Ph.D. in one of the social sciences, and among many things he does for a living is teaching a course in quantitative research methods at an Ivy League university. One of Bob's research interests is the connection between homelessness and foster care. There's research out there estimating that roughly 20 percent of youth that "age out" of foster care experience a significantly long spell of homelessness. Bob is convinced that this means that 20 percent of homeless people have aged out of foster care. I am not making this up; both myself and several other people have asked him to articulate his conviction as clearly as possible, and it is indeed how I've just described it. All attempts to show him the absurdity of said conviction have failed miserably.

Think about what this logic implies. Suppose 5 percent of smokers get lung cancer; this means you should believe that 5 percent of people with lung cancer are smokers. If 95 percent of men have a high school diploma, then 95 percent of people with a high school diploma are men. If half of the cats are black then half of all black things are black cats. Etc., etc. I think the sheer stupidity of this particular non sequitur is clear enough by now.

There are errors in reasoning that are subtle: it's easy to make them if one is not careful enough, and even if avoided, it's easy to understand where the mistake originates. None of the non sequiturs of the month fall into that category. Try as I might, I cannot come up with a plausible line of reasoning that could lead to them.