What is the probability given two randomly directed unit vectors $x$, $x'$ in the nth dimension, that $x'$ will be close to the origin than $x$. Given that $x'$ begins where $x$ ends.. I am trying to find an expression generalized with the dimension n for the probability that $x'$ will be closer to the origin than $x$ is.
For example in the first dimension, on a single axis. $x$ can either be 1 or -1, given an equally random probability to go in either direction. $x'$ can then either start at 1, or -1. If it starts at 1, it could go to 2 or back towards 0. Similarly for -1, $x'$ can go to -2 or back towards 0. This gives a $0.5$ probability that $x'$ is closer to the origin than $x$ is in the 1st dimensions.
For the second dimension, the step that $x$ will take can land anywhere on the unit circle $a$ centered at the origin. Since $x'$ will start somewhere on $a$ and move in any direction it could land anywhere on the unit circle $b$ which has it's center on any point of $a$'s circumference. This means finding the probability that $x'$ is closer to the origin is $x$ is equivalent to asking what portion of $b$'s circumference is inside of $a$'s. This works out to be $.333\dots$$ for the 2nd dimension.
I do not know the proof in the 3rd dimension to show how much of a spheres SA would be within the others if both touched each others mid points but from a question I have asked on this Mathematics StackExchange before I found out it was $.25$ which equates to the probability I am trying to find in the 3rd dimension.
the trend among the first three dimensions leads to guess of $1/(1 + n)$ as a generalized expression for the probability. Where n is the dimension. Is there a way to check or prove/disprove this for each higher dimensions?