I'm trying to understand a cryptanalysis of the Blowfish cipher, and I need to calculate the probability of collision in the cipher's S-boxes. Basically an S-box is a list of 256 semi-random 32-bit numbers. We can assume that the values are random for convenience. I need to know how I calculate the probability that
$$ S(a) = S(a') $$$$ a \neq a' $$
where a and a' are positions in the S-box.
After thinking about it for some time it seems the problem is trivial: The probability is just $\frac{1}{2^{32}}$.
If we're modelling S boxes as random functions from 8 bit numbers (but this doesn't even matter) to 32 bit numbers, then the chance of $a'$ (a number distinct from $a$) having the same $S$ value (which is just a random 32 bit number) as $a$ is just $\frac{1}{2^{32}}$.