C) Playing against a very rich player
(or a walker starting at $x_0 > 0$, in an open semi-infinite space, with no trap to the right
of him, just a trap at $x = 0$)
We showed that if q is bigger or equal than p, you will always lose with probability 1,
(no matter what is your initial $x_0$) but if q is smaller than p there is a probability of losing
given by $( rac{q}{p})^{x_0}$, which means that the probability of never coming back to the origin is
given by $1 - ( rac{q}{p})^{x_0}$
We also showed that in this last case it does not make sense to calculate the average
length of the game (since it could be infinity) but in the previous case (when you always
lose), you can calculate the length of the game and it will be:
(5)
$D_{x_0} = \frac{x_0}{q - p}$
Try to verify these two facts (probability of winning or losing when $q > p$, and length of
the game when $q < p$). You do not need to do it for so many values of $x_0$ as before, just
check it for three or four values.
NOTE: Understand that you cannot simulate an infinite system in the computer. What
you have to do is, for a certain value of $x_0$, to calculate these expressions for a large N (say
50, then 75, then 100, etc...) and see if you can plot the results and extrapolate for N
going to infinity. See how far you can go with this case.
