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How to solve it?

"Suppose that $N$ points are independently chosen at random inside a sphere of radius $R$. Find the probability of the distance between the center of the sphere and the closest point being greater than $r$, $r < R$, assuming the points are evenly distributed inside the sphere."

Gabriel Vivaldini
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1 Answers1

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What is the probability that a specific point is at distance larger than $r$ from the center?

It is clearly $1-(\frac{r}{R})^3$. You want the probability that this happens for all of the points.

Since the points are chosen independently the probability is $(1-(\frac{r}{R})^3)^N$

Asinomás
  • 105,651
  • How come? Is $(\frac{r}{R})^3$ the probability of the distance between a point and the center being less than $r$? Why? I don't understand your approach. – Gabriel Vivaldini Sep 24 '16 at 21:11
  • yes, that's exactly it. Why? The volume of the sphere of radius $r$ is $\frac{4\pi r^3}{3}$ and the volume of the sphere with radius $R$ is $\frac{4\pi r^3}{3}$. The ratio between the volumes if $(\frac{r}{R})^3$. So if you pick a point inside the big sphere, the probability that is inside the small sphere is $(\frac{r}{R})^3$ – Asinomás Sep 24 '16 at 21:13