Independence result where probabilistic intuition predicts the wrong answer?

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 $n$.