Independence result where probabilistic intuition predicts the wrong answer? 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.
 A: If the Borel-Cantelli lemma counts as probabilistic intuition, then here's an example.  Think of the real $x$ that you adjoin to a ground model as a sequence of $0$'s and $1$'s, and let $f(n)$ be the length of the $n$-th run of consecutive $1$'s in $x$. If the bits in $x$ were chosen by independent flips of a fair coin (or even of a biased coin as long as both sides of the coin have positive probability), then the inequality $f(n)>n$ would (with probability $1$) hold for only finitely many $n$ (by Borel-Cantelli).  But for a Cohen-generic $x$, that inequality holds for infinitely many $n$. In fact, for any function $g:\omega\to\omega$ in the ground model, $f(n)>g(n)$ for infinitely many $n$.
A: Freiling's Axiom of Symmetry (discussed also here) posits that for any function $f$ from $[0,1]$ to the at most countable subsets of $[0,1]$ there exists a pair of points $x$ and $y$ such that $x\notin f(y)$ and $y\notin f(x)$.  Freiling argued that, for a "random" $f$, $x$, and $y$, the probability that both $x\notin f(y)$ and $y\notin f(x)$ is 1.  However, the axiom is actually independent of ZFC (and actually equivalent to $\neg$CH).
