Tagged Questions
58
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
3
votes
2answers
336 views
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 ...
0
votes
1answer
97 views
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 ...
13
votes
3answers
296 views
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 ...
1
vote
2answers
155 views
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 ...
2
votes
1answer
99 views
Probabilities of Non-Regular Dice
Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die.
If I have an arbitrary 6-sided solid, how do you ...
3
votes
0answers
86 views
Computing the proportion of vectors with the same sign
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 ...
