Probability that distance of two random points within a sphere is less than a constant

Question

Two points are chosen at random within a sphere of radius $r$. How to calculate the probability that the distance of these two points is $< d$?

My first approach was to divide the volume of a sphere with radius d by a sphere with radius r. But it does not seem to work, at least not for $d > r$.

$$P = \frac{ 4/3 \pi d ^3}{ 4/3 \pi r ^ 3 }\tag{?}$$

Any help would be hugely appreciated.

What is $d$ here? Diameter? If so, then why do you refer to "radius d" and "at least not for for d > r"? — JRN, Aug 28 '15 at 14:13
d is the distance between the two random points and a constant — Erik Kalkoken, Aug 28 '15 at 14:15
Your first approach will not work when the first point is more than $r-d$ from the centre of the sphere — Henry, Aug 28 '15 at 14:21
To clarify lets assume the random function is an equal distribution of points within the sphere. — Erik Kalkoken, Aug 28 '15 at 17:18

score 3 · Accepted Answer · answered Aug 28 '15 at 14:41

3

How are you at three-dimensional integrals?
Let the first point be at $(0,0,z)$. There are two spheres: One of radius $r$, centered at the origin, and one of radius $d$, centered at $(0,0,z)$. I would let $r=1$ to remove one letter from your calculations.
The spheres' intersection is symmetric about the $z$ axis, which should help you calculate the volume of the intersection. You rotate the intersection of two semicircles about the $z$ axis.
Once you have $P(z,d)$, you need to calculate the following.
$$P(d)=\frac{\int_0^1 z^2P(z,d)dz}{\int_0^1z^2dz}$$

answered Aug 28 '15 at 14:41

Empy2

50,853

Thanks for your fast and comprehensive response. My math is limited, so I have problems to understand the solution. However I have 3 questions:
1. The first point is at random coordinates within the sphere with radius r, not at (0,0,z)
2. I am looking for a solution for different r, so setting it to 1 does not help me.
3. How do you define P(z,d) ??
What I am really looking for, is a function P(r, d), which gives me the %-probability for any r, d > 0. Ideally something that I can calculate in Google sheets.
– Erik Kalkoken Aug 28 '15 at 17:16
@bji74: $P(r,d)=P(1,d/r)$, so it suffices to solve the problem for $r=1$. – joriki Aug 28 '15 at 21:26
@bji74 (1) Pick the first point. Then change the coordinate system so that it is at (0,0,z). (2) P(r,d) = P(1, d/r) – Steven Alexis Gregory Aug 28 '15 at 22:44
@StevenGregory: I suspect you intended to ping bji74? – joriki Aug 28 '15 at 22:45

score 1 · Answer 2 · answered Aug 28 '15 at 21:48

1

According to MathWorld, the density for $d$ is

$$ \frac{3d^2}{r^3}-\frac{9d^3}{4r^4}+\frac{3d^5}{16r^6}\;. $$

Thus the cumulative distribution function that you're looking for is

$$ \int_0^d\left(\frac{3d'^2}{r^3}-\frac{9d'^3}{4r^4}+\frac{3d'^5}{16r^6}\right)\mathrm dd'=\left(\frac dr\right)^3-\frac9{16}\left(\frac dr\right)^4+\frac1{32}\left(\frac dr\right)^6\;. $$

Here's a plot.

answered Aug 28 '15 at 21:48

joriki

238,052

1

The above formula for the CDF is worked out in an answer to the following question: https://math.stackexchange.com/questions/3848889/probability-involving-two-points-inside-a-sphere question – Christian Blatter Oct 16 '20 at 10:48

Probability that distance of two random points within a sphere is less than a constant

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