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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?

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1 Answer 1

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Unfortunately your very plausible guess turns out to be wrong. You're looking for the fraction of the surface area of the sphere around $x$ that's within $\pi/3$ of the ray to the origin, and this can be computed as

$$ \frac{\displaystyle\int_0^{\pi/3}\sin^{n-2}\theta\,\mathrm d\theta}{\displaystyle\int_0^\pi\sin^{n-2}\theta\,\mathrm d\theta} $$

for $n\ge2$. For $n=2$ and $n=3$ this yields $1/3$ and $1/4$, respectively, as you found, but already for $n=4$ we have

$$ \frac{\displaystyle\int_0^{\pi/3}\sin^2\theta\,\mathrm d\theta}{\displaystyle\int_0^\pi\sin^2\theta\,\mathrm d\theta}=\frac13-\frac{\sqrt3}{4\pi}\approx0.1955\;, $$

almost but not quite $1/5$. You can compute these integrals recursively using integration by parts:

$$ \begin{align} \int_0^\alpha\sin^k\theta\,\mathrm d\theta &= \int_0^\alpha\sin\theta\sin^{k-1}\theta\,\mathrm d\theta \\ &= \left[-\cos\theta\sin^{k-1}\theta\right]_0^\alpha+(k-1)\int_0^\alpha\cos^2\theta\sin^{k-2}\theta\,\mathrm d\theta \\ &= -\cos\alpha\sin^{k-1}\alpha+(k-1)\int_0^\alpha(1-\sin^2\theta)\sin^{k-2}\theta\,\mathrm d\theta \end{align} $$

and thus

$$ \int_0^\alpha\sin^k\theta\,\mathrm d\theta =\frac1k\left( -\cos\alpha\sin^{k-1}\alpha+(k-1)\int_0^\alpha\sin^{k-2}\theta\,\mathrm d\theta\right)\;, $$

where the boundary term vanishes for $\alpha=\pi$ but not for $\alpha=\pi/3$.

For another problem involving spheres in $n$ dimensions in which similar integrals come up, see Puzzle: stars in the universe - probability of mutual nearest neighbors.

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