I read this problem a few days ago and just decided to give it some thought now (I would love to know the correct solution).
The answer will certainly be a function of n (because, for example, I could give you n = 1001 and that should have 0 probability whereas if I were to give you n = 1, that would have a very high probability).
My first thought was to write out a recurrence relation f of n and k (where n is the current level and k is the number of moves left). From this, we get: f(n, k) = .5f(n-1, k-1) + .5f(n+1, k-1) with f(n, k) = 0 when n > k and f(0, k) = 1. So to solve this, we just take f(n, 1000) = [a rapidly exploding function].
It appears that a basic knowledge of stochastic processes is required to actually be able to solve this problem, so it's one of those things that are actually much more complicated than they first appear.
The answer will certainly be a function of n (because, for example, I could give you n = 1001 and that should have 0 probability whereas if I were to give you n = 1, that would have a very high probability).
My first thought was to write out a recurrence relation f of n and k (where n is the current level and k is the number of moves left). From this, we get: f(n, k) = .5f(n-1, k-1) + .5f(n+1, k-1) with f(n, k) = 0 when n > k and f(0, k) = 1. So to solve this, we just take f(n, 1000) = [a rapidly exploding function].
Any thoughts?