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.

Geometric probability is a tool to deal with the problem of infinite outcomes by measuring the number of outcomes geometrically, in terms of length, area, or volume. In basic probability, we usually encounter problems that are "discrete" (e.g. the outcome of a dice roll; see probability by outcomes for more). However, some of the most interesting problems involve "continuous" variables (e.g., the arrival time of your bus

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Normal orthant probability in six variables

Let $\mathbf{X} \sim N(\mathbf{0}, \mathbf{\Sigma})$ be a $6$-dimensional Gaussian vector with covariance matrix of the form $$\mathbf{\Sigma} = \begin{pmatrix} 1 & c \\ c & 1 \end{pmatrix} \otimes \begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 1 & 1/2 \\…
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Average area of the shadow of a convex shape

What is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape? I know there are a lot of…
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What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?

The vertices are chosen completely randomly and all lie on the circumference. Is there a formula for the chance that an $n$-gon covers over $50$% of the area of the circle, with any input $n$? I tried to find something, however I did not know what…
volcanrb
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Probability of Intersecting Two Random Segments in a Circle

I designed this problem and tried to solve it but didn't solve. Choose four points $A$, $B$, $C$ and $D$ from inside of a circle uniformly and independent. What is the probability that $AC$ intersects $BD$?
user120269
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Random Triangle Inscribed in a Circular Sector

Lately, I have been thinking about expected area and perimeter of a triangle inscribed in a 'partial' circle or circular sector with radius $r$ and truth be told, I couldn't answer these questions. I hope someone here could give me a good answer.…
Tunk-Fey
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What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two points? (This is a generalization of this…
patricksurry
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Probability all angles of triangle formed within semircircle less than $120^\circ$

$3$ points $A$, $B$, $C$ are randomly chosen on the circumference of a circle. If $A$, $B$, $C$ all lie on a semicircle, then what is the probability that all of the angles of triangle $ABC$ are less than $120^\circ$? Okay, let's fix a semicircle…
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PDF of area of triangle with normally-distributed coordinates in any dimensions

Question What is the probability distribution function (PDF) of the absolute area of a triangle with normally-distributed coordinates in $\mathbb{R}^m$ $(m \in \mathbb{N}, m\ge2)$ ? A conjecture is given that can be proved or might help to find the…
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Probability that a Particle which moves Unit distance in a Random direction on each step will be inside the Unit Sphere after $n$ steps

The following integral equation arises while calculating the probability that, a particle which starts at the origin and moves a unit distance in a random direction on each ‘move’, will be within the unit sphere after $n$ moves: $$ f_n(x) =…
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Given a knight on an infinite chess board that moves randomly, what's the expected number of distinct squares it reaches in 50 moves?

I was asked this in an interview and wasn't sure how to frame the answer. Basically as in the question you have a knight on an infinite chess board and it chooses one of its valid 8 moves uniformly at each move. After 50 moves, the question was to…
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Continuity of the Euler characteristic with respect to the Hausdorff metric

Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means with respect to the Hausdorff distance: $$d_H(X,…
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Probability that the triangle is acute

A triangle is formed by randomly choosing three distinct points on the circumference of a circle and joining them. What is the probability that the formed triangle is an acute triangle?
Hussain-Alqatari
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How is the number of points in the convex hull of five random points distributed?

This is about another result that follows from the results on Sylvester's four-point problem and its generalizations; it's perhaps slightly less obvious than the other one I posted. Given a probability distribution in the plane, if we know the…
joriki
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How many squares can be made from points on $ z(t) = e^{2\pi i\, t} + \frac{1}{\sqrt{3}} e^{2\pi i\, 3t} $?

Inspire by the Toeplitz Square Problem, how many squares can be drawn on the curve: $$ z(t) = e^{2\pi i\, t} + \frac{1}{\sqrt{3}} e^{2\pi i\, 3t} $$ wth $t \in [0, 2\pi]$. Here is an image: We're up to one five nine ten squares. Here is an…
cactus314
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Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a disk) is $O(n^{1/3})$. These bounds also extend to…
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