You have been captured and blindedfolded by pirates, then placed on somewhere on a 5 meter long plank. You have 0.5 probability of moving 1 meter, and 0.5 probability of moving -1 meter. One end of the plank leads you to safety, while the other end leads to death. If $x$ (integer) is the distance in meter you start from the safe end, determine the probability of your survival as a function of $x$?
Attempt:
Simply write out the conditional probability. Let $p_1$ = probability of survival if started from x =1. Thus $p_1= 0.5 + 0.5 p_2$, and $p_2 = 0.5p_1 +0.5 p3....... $ And $p_4=0.5p3$. You can solve this system of equations.
However, the book also explained that there is a martingale approach. I am not sure whether this is correct.
Attempt 2 (Martingale):
$E[S_x] = 0 = (p_x * x + (1-p_x)*(5-x)) =0, $ this wil give you $p_x$ for probability of surival if start from x. However, I am not sure why $E[S_x]=0.$
The $S_x$ in the Martingale example is your net displacement from your starting point at the time when you exit the plank (whether it bad the bad way or the good way). i.e $x$ is a random variable representing the time you leave the plank.
The intuition is that because $S_n$ is a martingale $E(S_n)$ is going to be equal to $0$ at any time $n$, thus why not at time $x$?
The complication, is that $x$ is a random variable, so the reasoning above does not go through, and in fact there are many situations where you cannot draw the conclusion. However, in many cases you can, (see optional stopping theorem) and this is one of them, since the Martingale is bounded and $x$ is almost surely finite.
So you have $E(S_x)=0$ and the equation you wrote down can be solved for the survival probability.