Since the other thread wants it merged into here, I'll post here instead. Because I'm kind like that.
As I said in the other thread, the maths does work for improving the stats in a seemingly (at first glance) entirely random set of events.
Its easiest explained using the examples that were used in the program:
Player 1 picks HHH
Player 2 flips the middle result (HTH) and moves to the front (THH).
The first result, lets say its heads. Its a 1/2 chance of getting heads.
The second result, lets say it too is heads. Its also a 1/2 chance of getting heads, but a 1/4 chance of player 1 getting a HH combination.
The third result is where the stats work in your favour.
For player 1 to complete their set of HHH, they have a total 1/8 chance. However, as player 2 at this point you need to break their combination, by getting a tails. This is not affected by prior results, so your chance of breaking their combo is 1/2. The odds are in your favour to break the combination (1/2 against 1/8). Additionally, this also BEGINS your combination. So if the next 2 results are HH you get a point. If in the next 2, one of the results is T then noone gets a point (but you begin your run again).
At any given point, the odds of you getting a point are at least equal to the other player.
Thats a quick explanation. If I was unclear, and people would like a better explanation, say and I will try to oblige