I think that it is better to think about it in terms of the 'colour of bits'. Bit colour can be thought of as an invisible property that follows a bit around whenever that bit is computed in a way that depends on another coloured bit.
Suppose n_1 is picked randomly (from a distribution that doesn't take into account any 'coloured' bits). Suppose n_2 is computed from illegally obtained data. Now suppose that numerically, n_1 == n_2. Mathematically, n_1 and n_2 are indistinguishable, but in the eyes of the law, n_1 is not coloured and therefore legal, while n_2 is illegal.
In practice, courts cannot see the colour of bits, so they have to infer the colour. If you claim that some data on your hard disk was randomly generated, but it is exactly the same as, say, a 100 GB copyrighted movie file. The probability of randomly generating such a small file (unless your distribution is stacked towards that file - which doesn't count as being uncoloured) is so small that it is not credible that you coincidentally generated it randomly, so the court will likely infer that the bits in that file are coloured.
Another more complex example: suppose I give you four bitstrings, A, B, C, and D. Let ^ represent bitwise XOR. Suppose that A is a representation of the complete public domain works of Arthur Conan Doyle, A^B is a public domain representation of War & Peace, B^C is a copyrighted novel (and the person distributing the strings doesn't have permission to distribute it). C^D is a public domain representation of the bible. Which bitstrings are illegal? To decide, a court will need to infer how you generated the strings. A is clearly not illegal because it matches a public domain work exactly. B is completely determined by A and the contents of War & Peace. To compute C, you would take a copyrighted work and xor it with B, so that makes C illegal to distribute without permission because it was derived from a copyrighted work. To compute D, you need C and the bible, and since C is coloured, D is also coloured.
Suppose n_1 is picked randomly (from a distribution that doesn't take into account any 'coloured' bits). Suppose n_2 is computed from illegally obtained data. Now suppose that numerically, n_1 == n_2. Mathematically, n_1 and n_2 are indistinguishable, but in the eyes of the law, n_1 is not coloured and therefore legal, while n_2 is illegal.
In practice, courts cannot see the colour of bits, so they have to infer the colour. If you claim that some data on your hard disk was randomly generated, but it is exactly the same as, say, a 100 GB copyrighted movie file. The probability of randomly generating such a small file (unless your distribution is stacked towards that file - which doesn't count as being uncoloured) is so small that it is not credible that you coincidentally generated it randomly, so the court will likely infer that the bits in that file are coloured.
Another more complex example: suppose I give you four bitstrings, A, B, C, and D. Let ^ represent bitwise XOR. Suppose that A is a representation of the complete public domain works of Arthur Conan Doyle, A^B is a public domain representation of War & Peace, B^C is a copyrighted novel (and the person distributing the strings doesn't have permission to distribute it). C^D is a public domain representation of the bible. Which bitstrings are illegal? To decide, a court will need to infer how you generated the strings. A is clearly not illegal because it matches a public domain work exactly. B is completely determined by A and the contents of War & Peace. To compute C, you would take a copyrighted work and xor it with B, so that makes C illegal to distribute without permission because it was derived from a copyrighted work. To compute D, you need C and the bible, and since C is coloured, D is also coloured.