## Problem understanding a fact

Hello ! I have troubles understanding this statement :

If one, in an attempt to factorize a composite number $n$, finds two squares $a$ and $b$ at random such as $a^2 - b^2 \equiv 0 \pmod{n}$, then taking the greatest common divisor of $a \pm b$ with $n$ yields a nontrivial factor of n with probability $\frac{1}{2}$.
I don't understand why there is 0.5 probability to find a nontrivial factor of $n$, since if we call $p$ one factor of $n$, then if we find a nontrivial factor, $p$ must divide $a \pm b$, and so there are $\frac{n}{p}$ possibilities that this happens, whether if we find a trivial factor of $n$, then $n \ | \ a \pm b$, and there are infinitely many numbers $a$ and $b$ that satisfy this condition. So it should be very likely that we find a trivial factor of $n$ with a difference of squares, which is not true.

Can somebody explain to me where I went wrong in my reasoning ? Thanks !