If you have a file and XOR every bit with a random bit, can you extract any information? Say you have a file that is not random, and you XOR every bit with a random bit (not pseudo, but really random). Can someone who sees only the result extract any information from it? Obviously, it won't be 100% accurate, but I imagine you can do some sort of stochastics and get a vague idea. If yes, how? If no, is there a mathematical proof?
This cipher is called a one-time pad. It is unbreakable ("perfect secrecy") assuming that:


The pad (the collection of random bits) really is truly random
The pad is never reused to encrypt other messages


So, no information can be extracted from $\text{file} \oplus \text{random bits}$. 

The basic idea of the proof is that an attacker can test every possible key, but they have no way of knowing which plaintext is actually correct. If I encrypt "attack" with a one-time pad, then any six-character string could just as equally have been encrypted in the first place.