In some of his writings, Paul Cohen gave an informal, motivational discussion about the word generic (as it is used in forcing). While very suggestive, the discussion leaves the meaning of the word ambiguous, and could lead someone to guess that it is related to probability theory. Indeed, there was a recent MO question along these lines. As the comments and answers to that question make clear, generic is actually closely related to Baire category and not to measure theory. As one learns in an analysis course, it is perfectly possible for a comeager set to have measure zero and for a meager set to have positive measure.
This got me wondering. Is there a simple/natural example of an independence result where, if you were to appeal to probabilistic intuition, you would guess wrongly what happens, because the generic event has probability zero? A good example should (A) be an independence result that is natural-sounding and easy for a non-set theorist to understand and (B) have an obvious and tempting—but wrong—line of reasoning based on conflating measure with category.