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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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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