What is the probability of a random string of numbers having a consistent pattern?

I saw a video by Michael Shermer today, where he said people who believed in ESP etc. were more likely to see patterns in selected pictures than the average person, whether or not the picture held a pattern.

This got me thinking, what is the probability of a random data set having a pattern? Of course with pictures it would be astronomically high without a lot of approximation, but what about a simple string of numbers? Is there a rule that can describe this?

Here is a random string I have generated as an example:
8177322396

Within this, if I had only produced 3223, it could be an inflection, or a repeated pattern 2233… If I only had 396, it could be 3,9,6,12,9,15… If I only had 732, it could be the end of the pattern 32,16,7,3,2. As a whole though, the string has no pattern that I can see. The probability of a pattern therefore decreases with string length, but is there some sort of relationship that we can describe empirically?