The following problem has 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 in the following way. For every step of the "walk", it will choose a real number $\Delta x$ uniformly from the interval $[-1,1]$, turn right, and move $\Delta x$ unit. Once it reaches the left side of the point $O$, it will "die" immediately.

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