I have seen this short, wonderful video about the frequency stability property (also there is another video from this guy under his Kickstarter project ArtOfTheProblem; they are both great, and I highly recommend them).
It basically said that when you have two sequences of 0s and 1s you can determine which of those two sequences was truly generated randomly (using a coin flip procedure) and which one was generated by a human trying to act randomly. The trick to discern the two sequences from one another starts with counting non-overlapping segments of length 3. For any sequence you pick, the proportion of times it occurs should be the same as all the other sequences.
Consider following example
1st: Which 3 numbers should we observe? Or in another words, should the window of size 3 overlap? Is this the correct histogram?
2 x 101 1 x 010 1 x 100 1 x 000 1 x 001 1 x 011 1 x 110
Or if the window should not overlap then what to do with last 1:
1x 101 1x 000 1x 110 1??
2nd: Why should be observer 3 subsequent numbers? Does it have something to do that the string consist of two numbers (0 and 1)? What would happen if we observe 4,5,6 ... subsequent numbers?
3rd: Do you know some other channel on youtube like this one?
EDIT: I've done experiments using (non)overlapping window, results are here