Questions tagged [geometric-probability]
Probabilities of random geometric objects having certain properties (enclosing the origin, having an acute angle,...); expected counts, areas, ... of random geometric objects. For questions about the geometric distribution, use (probability-distributions) instead.
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Probability of 3 darts landing in the same half of the board [duplicate]
Problem: Find the probability of 3 randomly thrown darts landing in the same half of the board.
More generally, if $n$ points picked uniformly randomly on a disk, find the probability of them lying in ...
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Kolmogorov bound for comparison of Random Vector Projections on a Sphere [closed]
Let $n$ be a fixed integer and $X$ be a random vector in $\sqrt{n} S^{n-1}$ (the $\sqrt n$-radius sphere in $\mathbf{R}^{n}$) with a density $f$ which satisfies the following property:
$
\forall x \in ...
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Probability distribution for the perimeter of a random triangle in a circle
This is the same question as this question except that the random triangles do not need to necessarily touch the circle and I am only interested in the distribution for the perimeter. And I am ...
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Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and the $x$-axis, from $x=0$ to $x=\pi$.
Where the question came ...
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Is there a formal proof that points taken at random in a bound area are evenly distributed?
I am an amateur trying to understand how probability works on the euclidean plane.
Despite my efforts I couldn't find any formal proof that points taken at random in a bound area are evenly ...
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How can I find the average distance between two points inside a torus
Suppose we have the torus with equation $1-\left(\sqrt{x^2+y^2}-3\right)^2=z^2$. We choose two randomly chosen points inside it with a uniform distribution. I want to find the average distance between ...
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Average distance between random points inside a semisphere and a quarter-sphere
Suppose we have a semisphere of radius 1. We choose two random points inside it with a uniform distribution. That is, if we pick random points insed it, they will be uniformly distributed.
What is ...
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Distribution of distances between two randomly selected points in a semicircle
Suppose we have a semicircle with radius $1$: We choose two random points with a uniform distribution, that is, that if we pick random points inside it, they will ...
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Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.
My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation?
Context
Some probability questions have answer $\frac{1}{2}$, and - as ...
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A mysterious limit: probability that a triangle captures the centre of a circle.
On a circle, choose $6n$ $(n\in\mathbb{Z^+})$ uniformly random points and label them $a_0,a_1,a_2,\dots,a_{6n-1}$ going anticlockwise, with $a_0$ chosen randomly.
Draw three chords:
Chord $a_0 a_{3n}$...
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A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. The petals are randomly chosen, but they must be ...
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Expected number and variance of number when sum exceeds 1 (Irwin-Hall distribution)
I am given a random variable (uniformly distributed) between 0 and 1. To this, I add a second such random variable. I keep on adding these variables until sum exceeds 1, and then stop. Let us call ...
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Boyd & Vandenberghe, Exercise 2.15g — Convexity of a region on the probability simplex
Exercise 2.15g of Boyd & Vandenberghe's Convex Optimization:
On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1, p_2, p_3, \ldots, p_n)$ corresponds to a distribution for ...
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The vertices of a hexagon are random points on a unit circle; $a,b,c$ are the lengths of three random sides. Conjecture: $P(ab<c)=\frac35$.
The vertices of a hexagon are uniformly random points on a unit circle; $a,b,c$ are the lengths of three distinct random sides.
A simulation with $10^7$ such random hexagons yielded a proportion of $0....
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Is there anyway to guarantee probability mass coverage?
If I have a probability density function on $\mathbb{R}^n$. I can sample $m$ points from it. Is there anyway to get an estimate of how much probability mass is covered by balls of radius $\delta$ ...
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