The following problem bothered me for a long time.

Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly. For every step of the "walk", it will choose a real number $\Delta x$ in interval $[-1,1]$ equiprobably, and turn right and move $\Delta x$ unit. Once it move to the left side of the point $O$, it will "die" immediately.

Our task is find out the probability of the point "live" after $n$ steps of "walk" $P_n$. I guess that $P_n=\frac{(2n)!}{4^n (n!)^2}$. But I can't prove that it is correct or explain why.