If you take this sort-of-stretched case where encryption can be broken, I can explain why this is a problem.
Suppose you are using the PRNG to make a stream cipher. Basically, your random key is a seed for the PRNG. You then generate lots of pseudo-random characters from that seed, and XOR them together (character-by-character) with your message to encrypt it.
Now, the fact that XOR is linear (it's just addition mod 2) means when you XOR two probability distributions against a constant (i.e., an atomic distribution), you'll get a shifted distribution. Let's say your PRNG disproportionately outputs "0" at each character. Then the distribution of each character of the ciphertext will be centered at p XOR 0, where p is the corresponding character of the plaintext!
So by the law of large numbers, if we see the same message encrypted many times, we can determine with high probability exactly what each character of the plaintext is, and completely break encryption!
Suppose you are using the PRNG to make a stream cipher. Basically, your random key is a seed for the PRNG. You then generate lots of pseudo-random characters from that seed, and XOR them together (character-by-character) with your message to encrypt it.
Now, the fact that XOR is linear (it's just addition mod 2) means when you XOR two probability distributions against a constant (i.e., an atomic distribution), you'll get a shifted distribution. Let's say your PRNG disproportionately outputs "0" at each character. Then the distribution of each character of the ciphertext will be centered at p XOR 0, where p is the corresponding character of the plaintext!
So by the law of large numbers, if we see the same message encrypted many times, we can determine with high probability exactly what each character of the plaintext is, and completely break encryption!