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 ...
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Expected radius of throwing a dart at a dartboard
I am doing a problem that states: If you are throwing a dart at a circular board with radius $R$, what is the expected distance from the centre?
If $x$ is the expected radius, then it would be the ...
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Multivariate Normal Distributions and the Uniform Distribution on the Sphere
Given a multivariate normal vector $X \sim N(0,I_d)$ (identity covariance matrix), it is well known that :$$\frac{X}{\|X\|_2}
$$
is uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb{...
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Find the best ratio between grid size and the square size
Let's say I have a bunch of squares of side x and the grid of square sectors, each of side y.
I am placing the squares randomly in this grid - the sides of squares are parallel to sides of the grid, ...
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Expected area of inscribed triangle
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that formula: $S = \frac{1}{...
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Distribution of a combination of four uniformly distributed variables: $ X_1+X_2 +\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$
My problem involves four random variables $X_1, Y_1, X_2, Y_2 \sim U(0,1)$ in the expression $Z = X_1 + X_2 + \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$. From what I understand so far, I need to find the ...
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$n\times n$ grid filled with $n$ colors. What is the average group size as $n\to\infty$
Take a grid with dimensions $n\times n$ squares and randomly fill each square with $1$ of $n$ colors. What is the expected average group size of colors touching each other as $n$ approaches $\infty$? ...
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Three random points on $x^2+y^2=1$ are the vertices of a triangle. Is the probability that $(0,0)$ is inside the triangle's incircle exactly $0.13$?
Three uniformly random points on the circle $x^2+y^2=1$ are the vertices of a triangle.
What is the probability that $(0,0)$ is inside the triangle's incircle?
(This a variation of the question &...
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A square contains many random points. From each point, a disc grows until it hits another disc. What proportion of the square is covered by the discs?
A square lamina contains $n$ independent uniformly random points. At a given time, each point becomes the centre of a disc whose radius grows from $0$, at say $1$ cm per second, and stops growing when ...
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How to define pdf of the distance to the point of the spherical cap?
Suppose we have a sphere centered at $(0, 0, 0)$, with the radius of $R_b$. We cut the sphere with the tangent plane centered at $(0, 0, R_a)$, where a dude is fixed on. (Here $0 < R_a < R_b$, ...
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A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?
A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
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Break a stick at two random points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/2$. Why?
Choose two independent uniformly random points on a stick, and break the stick at that those points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/...
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What is the probability that the triangle formed by three uniformly random points on the sphere contains its circumcentre?
In answering Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$. it turned out that if you choose three ...
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The probability of getting exactly $k$ crossings in buffons needle problem
I am studying Buffons needle problem and I am currently trying to derive the probability of getting exactly $k$ crossings for the situation $l > d$ where $l$ is the needle length and $d$ is the ...
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Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$.
On (not in) a sphere, choose three independent uniformly random points $A,B,C$. Is the following conjecture true:
The expectation of the ratio of the area of (planar) $\triangle ABC$ to the area of ...
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Cut a unit stick at $n-1$ random points. Expectation of product of fragment lengths is $\prod\limits_{k=n}^{2n-1}\frac1k$. Why?
On a straight stick of length $1$, choose $n-1$ independent uniformly random points. Cut the stick at those points, yielding $n$ fragments.
Let $\mathbb{E}_n$ be the expectation of the product of ...
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Sufficient condition for 3 positive scales to represent sides of a triangle [duplicate]
Now with the above condition check the following video at 4:00
Check https://youtu.be/Zlnpo3GxWpY?si=Pagh6pcxgmRmEUHy
At 4:00
Any one of the inequalities (written with red color) should be sufficient ...
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If $(a,b,c)$ are the sides of a triangle, what is the probability that $ac>b^2$?
Let $a \le b \le c$ be the sides of a triangle inscribed inside a fixed circle such that the vertices of the triangle are distributed uniformly on the circumference.
Question 1: Is it true that the ...
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Probability for the highest correlation between random vectors
Let $k_1,\dots,k_m\in\mathbb{R}^d$, denote by $\mathcal{D}:=\mathcal{N}\left(0,\frac{1}{d}I_d\right)$ and $[n]= \{ 1,\dots,n \} $. I am interested in the following probability:
$P_{ x_1,\dots,x_n\sim \...
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Buffon's needle in one dimension
I want to solve Buffon's Needle problem but first I was trying to tackle a simpler case.
So: consider an infinite line with points each $t$ units. Let's say that we have a "needle" of length ...
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Probability that two random lines intersect inside a square
Consider the square with vertices $(0,0),(1,0),(1,1),(0,1)$.
Choose two independent uniformly random points $P$ and $Q$ inside the square.
Draw a line $l_P$ connecting $(0,0)$ and $P$.
Draw another ...
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