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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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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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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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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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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Probability between a rectangular 2D die and a squared 2D die
I'm trying to find a solution to the next problem:
If I have a rectangular $2D$-die with uniform density such that each side has a certain probability $P1,P2,P3,P4$ respectively.
I want to find a ...
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A probability involving side lengths of a random triangle on a disk: Is it really $\frac37$?
Choose three uniformly random points on a disk, and let them be the vertices of a triangle. Call the side lengths, in random order, $a,b,c$.
What is $P(a^2<bc)$ ?
A simulation with $10^7$ such ...
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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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The vertices of a pentagram are five random points on a circle. Conjecture: The probability that the pentagram contains the circle's centre is $3/8$.
The vertices of a pentagram are five uniformly random points on a circle.
Is the following conjecture true: The probability that the pentagram contains the circle's centre is $\frac38$.
(The ...
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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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Measure transport by a random matrix
I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are ...
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Probability that Mercury is the nearest planet to Earth.
Motivation: We tend to think of Venus as the nearest planet to Earth because at its nearest approach to Earth, Venus is the closest at 39 million Km away. This is followed by Mars at 56 million Km and ...
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Probability of a random cyclic quadrilateral enclosing a fixed point in its circle
I finally found a single integral solving the natural generalisation of the problem discussed here:
For $n\ge1$ pick $n+2$ points uniformly at random on the unit circle. What is the probability $P_n(...
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Probability of two geometric conditions happening together
I have a problem combining geometry with probability, and I feel like I do not understand the basics to approach this problem.
Let there be two points $\mathbf{p}_1$ and $\mathbf{p}_2$ in $\mathbb{R}^...
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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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Conjecture: Two different random triangles (both based on random points on a circle) have the same distribution of side length ratios.
On a circle, choose three uniformly random points $A,B,C$.
Triangle $T_1$ has vertices $A,B,C$. The side lengths of $T_1$ are, in random order, $a,b,c$.
Triangle $T_2$ is formed by drawing tangents to ...
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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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Draw tangents at 3 random points on a circle to form a triangle. Show that the probability that a random side is shorter than the diameter is $1/2$.
Choose three uniformly random points on a circle, and draw tangents to the circle at those points to form a triangle. (The triangle may or may not contain the circle.) For example:
What is the ...
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Probability that the centroid of a triangle is inside its incircle
Question: The vertices of triangles are uniformly distributed on the circumference of a circle. What is the probability that the centroid is inside the incricle.
Simulations with $10^{10}$ trails ...
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The probability of a circle in a circumscriptible polygon
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
why "the area of the polygon $abcelef$
.... represents the number of ways the three ...
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Probability of each type of inscribed octahedron
Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. ...
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Probability that the coefficients of a quadratic equation with real roots form a triangle
Question: What is the probability that the coefficients of a quadratic equation form the sides of triangle given that it has real roots? Assume that the coefficients are uniformly distributed and ...
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The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.
The vertices of a triangle are three uniformly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
Show that $P(ab>c)=\frac12$.
The result is strongly suggested by ...
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Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square
Question: Suppose that points $P_1$, $P_2$, and $P_3$ are chosen uniformly at random on the sides of a square $T$. Compute the probability that $$\frac{[\triangle P_1 P_2 P_3]}{[T]}>\frac{1}{4}$$ ...
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Probability Theory: Generating Functions of Random Variables
Let $X, Y$ be independent random variables with the geometric distribution with parameter
$p > 0$.
(a) Compute the mean of $Z = XY$.
I got that $E(Z) = 1/p^2$
(b) Compute the probability ...
