I'm reading Fractals, Chaos, Power Laws by Schroeder right now. In one chapter, he mentions this:
One of the several counterintuitive facts of the fair-coin tossing game is that the expected number of times equals 1 that a player will increase his capital by any given amount G before the first return to his initial capital -- independent of how large the gain G is!
A similarly counterintuitive result concerns the duration $D_k$ of the fair game, obtained by solving a difference equation similar to equation 7: $D_k = K(B-K)$
In other words, if the player has just a single dollar and the bank has $1 million, the expected duration of the game is 999,999 tosses!
I'm having trouble understanding why the duration is so long. Intuitively, it seems like it should be shorter. I ran a simple simulation of this game, and over 10,000 simulations, the average duration is 6,548 flips. What am I missing?
I also don't understand the justification for the provided difference equation. Assuming K is the number of dollars the player has, and B the number of dollars the bank has, then I don't get the equation. Doesn't it imply that the length of a game where the player and bank have the same amount should last 0 flips ($(B-K) = 0$)? Am I just wildly forgetting how difference equations work?
I apologize if I'm missing something obvious, but I haven't been able to find an explanation for this result from looking at various pages on the gambler's ruin problem.