You have been captured and blindfolded by pirates, then places somewhere on a five-meter-long wooden plank. Disoeriented, each step you take is a meter long but in a random direction – either towards the sharks waiting at one end or eventual freedom at the other. If x (integer) is the distance in meters you start from the safe end, determine the probability of your survival as a function of x.
My attempt, $M_n$ be position at time $n$, so that $M_0 = x$ and clearly is a martingale.
Let $\tau = min\{n: M_n = 0 \space\text{and} \space M_{n-1},…,M_0 \neq 5\}$. (Case $M_0 = 5$ is trivial.)
Then by optional stopping theorem we have $\mathbb{E}M_{\tau} = \mathbb{E}M_0 = x = 0 * \mathbb{P}(\text{reaching safety before sharks})$
But clearly I'm not able to get the probability out of this.
Could please someone explain what is wrong?
Best Answer
What you really want is the stopping time $T=\min\{n\mid M_n=0\text{ or }M_n=5\}$. This stopping time is well-defined everywhere.
After that, you had the right idea... indeed, we have $E(M_T)=E(M_0)=x$. And normally you'd be right, in general the expected value of a random variable tells you almost nothing about the probability distribution of the variable. But this is a very special case, and it happens that in this case, you can in fact deduce the probability you want from the expected value.
Here's a hint: what values can the variable $M_T$ take?