understanding a gambling strategy diagram

Question

I find myself spending a lot of unnecessary time and effort trying to understanding tables in probabilities that authors are attempting to illustrating - very frustrating

In the last diagram in the below example, I am unable to understand the "structure" nor the meaning behind it.

If L is the time of the last win during 6 games being played and N is the number of win in the first 6 games, L=1 and N=0 doesn't make any logical sense.

Any help is appreciated.

Edit: I am closing to the end of Martingale and, by far, one of the worst book. I do not recommend. Essential of Stochastic Process

Answer 1

You will always win $1$ on a chain of bets that ends in a win for you. The problem comes if you can't complete the chain. That can happen because you run out of money or because your opponent refuses to accept the next bet. If there is a house limit of 64 you can only make seven bets. $\frac {127}{128}$ of the time you win $1$. $\frac 1{128}$ of the time, when you lose all seven bets, you lose $127$. The net of this is $0$, as it must be if the game is fair.