# Tagged Questions

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### Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ ...
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### Probability that points are on a straight line

I am looking at a formula to calculate the probability that $n$ points are on a straight line between point $1$ and point $n$ in 2d Euclidean space. If the points are exactly on the line, the ...
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### Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
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### Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
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### A question about a metric space

I have a question regarding the following problem. Let $A,B,C$ be the three independently selected, uniformly distributed points on the unit sphere $S^3$ in $\Bbb R^4$. What's the probability ...
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### What is the expected area of a polygon whose vertices lie on a circle?

I came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly ...
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### Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...