[bhauer]

I am out of my element here, but my understanding is that since the key is equal in length to the message, there is no way for you to know whether you are simply seeing a pattern in the key or a pattern in the message.

Imagine a one time pad made for encoding numbers that used a "MOD 10" operation on each digit.

Then imagine the key is:

    6926560279774

And the message is:

    0000000000000

The output is:

    6926560279774

Alternative messages:

    1234567890123 -> 7150027069897
    1111111111111 -> 7037671370885

In all cases, the patterns that you can discern may be from my message and may be from the key. As an analyst, you can't tell.

If this were English letters rather than numbers, and you know 'e' is very common, you still can't get anywhere because each 'e' is encoded with a unique character from the key.

[benchaney]

This is a good description, but to add on to it: If there is a pattern in the plaintext, it does not increase the probability that there is a pattern in the ciphertext. It is true that there may be patterns in the ciphertext, but they give you no information about if there is a pattern in the plaintext.