# What is the probability of a unit vector composite being closer to the origin than one vector in the nth dimension

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.

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