I've been wondering about this for awhile now: does random data compress?

Some people seem to think it does, while others maintain that it doesn't. I mean to mathematically prove that only 1/255 of random files can compress.

Conditions:

Original file size: 32 bytes

Maximum size of "compressed" file: 31 bytes

Number of possible different original files: 1.158 * 10^77

Number of possilbe "compressed" files: let's calculate that

Uhoh, this is gonna be hard to do without summation sybols.... well, ASCII art never hurt anyone.

31_

\

/__ 1.158 * 10^77 * (1/256)^k = 4.540 * 10^74

k=1

(1.158 * 10^77) / (4.540 * 10^74) ~ 1/255

Does this or does this not prove that only 1/255 of random data files can be compressed, and is my math correct?