You have been captured and blindedfolded by pirates, then placed 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 $x \in {0,1,2,3,4,5}$ is the distance in meter you start from the safe end, determine the probability of your survival as a function of $x$?
My attempt (Martingale):
$E[S_x]=0=(p_x x+(1−p_x)(5−x))=0$, this wil give you $p_x$ for probability of survival if start from $x$.
But when I put in let's say $x=1$, I get the that $p_x=4/3$. What is wrong with my approach as a probability can't be over 1.
We know (see this) that for a random walk starting at 0 the probability to hit a boundary $b$ before boundary $-a$ ($a, b \in \mathbb{N}$) is $$\frac{a}{a+b}.$$ Using this result the calculations should not be too hard. For example, if $x=1$ we know that you die with probability $$\frac{1}{1+4}.$$ You should be able to figure out the general formula for $x$ from this.