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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4
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1answer
917 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
4
votes
1answer
126 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
7
votes
1answer
155 views

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 ...
0
votes
2answers
147 views

random circle with radius r on cartesian plane, probability of it not cutting x and y axis with intercepts.

I have a tough question here. Choose a circular disk of radius r on the cartesian plane. What's the probability it is not cut by horizontal lines with integer y intercept, or vertical lines with ...
2
votes
1answer
103 views

Convexity of a region on probability simplex

Exercise 2.15 g of Boyd et al Convex Optimization book : 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 random variable ...
5
votes
2answers
524 views

What is the average length of 2 points on a circle, with generalizations

I have earlier seen the question about finding the average length of two points and $n$ points inside the unit disk. But what about the more simple question, what happens if the points lie exactly on ...
2
votes
1answer
147 views

Expected value of maximum distance between points

Consider a two dimensional square domain ($S$) of size $l \times l$. We generate a point $\mathbf{x}_i = (x_i,y_i)$ in S with uniform distribution, i.e., the point is equally likely to be anywhere ...
2
votes
1answer
176 views

Three random points on a circle, CDF for second largest angle

Three points $A$, $B$, $C$ are chosen randomly on a circle. Let us consider angles $\alpha$, $\beta$, $\gamma \in [0, 2\pi)$ formed by consecutive pairs of points. Angles are reordered from the ...
13
votes
1answer
158 views

Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what ...
1
vote
1answer
168 views

Geometric Probability- Circle and two points

A point $P$ is chosen $0.5$ units away from the centre of a circle of diameter $2$. Now two points are chosen randomly on the circumference of the circle. What is the probability that the triangle ...
1
vote
2answers
82 views

Integral of convex set

Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely, why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
3
votes
1answer
83 views

Motivation and application for stochastic geometry.

I am starting a PhD, and there is a good chance that my project will be oriented in the study of random polytopes or/and random mosaics. I was wondering what are the motivations and applications of ...
2
votes
0answers
81 views

Packing a larger sphere with smaller spheres in high dimensions

We were discussing today the probability of leaving a point uncovered while trying to fill a larger sphere by randomly throwing in smaller spheres. Here's the argument: We are working in ...
2
votes
0answers
87 views

Expectation number of random points exactly on their convex hull

Suppose there are n random points uniformly distributed in a square, what's the expectation of the number of the points located exactly on the edge (or being vertexes) of their convex hull? What if ...
0
votes
1answer
614 views

Probability that coin will fall into a square

So the exercise is this: We have and infinite chessboard and we have a coin. Every grid is of length and width $a$, whereas the coin has diameter $2 \cdot r<a$. We throw a coin into a chessboard ...
3
votes
0answers
45 views

Random convex shapes containing a ball

I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space. Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
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vote
2answers
424 views

Probability that centre of the square lies inside the circle joining the two points inside the square

Two points are uniformly and independently distributed (located) inside a square. A circle is drawn such that the segment joining the two points is a diameter. Find the probability that the center of ...
1
vote
1answer
246 views

Random points in a cube.

A point with coordinates $x$,$y$,$z$, is chosen uniformly at random from a cube: $$\{(x,y,z)\in \mathbb{R^3}:0\le x,y,z \le 10\}.$$ Assume that the probability of an event is ...
0
votes
0answers
83 views

Spatial distribution of bees

* Please please help! I still get stuck. We have a forest for bees, consisting of $4$ non-overlapping regions. $80\%$ of the bees seek honey in the forest while $20\%$ of the bees do so outside the ...
1
vote
1answer
228 views

Probability of finding a point on or in an $n$-dimensional unit sphere

If a point is chosen at random in an $N$-dimensional unit sphere, what is the probability of falling inside the sphere of radius $0.99999999$? What if $N=3$, $N=10^{23}$, or $N = \infty $? Okay, that ...
5
votes
2answers
106 views

finding the number of circles we get when randomly placing given patterns into a grid of squares

We have an 11$\times$11 table of squares (consist of 121 squares of dimension 1$\times$1). we have 3 tiles shown in the picture. Each tile has dimension 1$\times$1. we now randomly pick 3 tiles into ...
3
votes
1answer
181 views

Truchet tiles on a flattened cube

We have 2 Truchet tiles and a flattened cube as shown. We randomly place copies of the tiles into faces of the flattened cube. Find the probability that the circular arcs on the Truchet tiles ...
0
votes
2answers
187 views

Geometric probability question

So there are 2 parallel lines 20 feet apart. A piece of pipe 20 feet long falls between the lines and one end is exactly 10 feet from one line. What is the probability that the pipe lies entirely ...
1
vote
1answer
87 views

A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it. What is the probability that the needle lies between the two lines? I can see how ...
2
votes
2answers
106 views

Probability of line intersecting the convex set.

I would like to prove this theorem: Let $A,B \subseteq \mathbb{R} ^3$ be convex, limited sets. $B \subseteq A$. I have a "random line", which intersects A. Probability, that this line also intersects ...
2
votes
1answer
751 views

Expected area of the intersection of two and three circles

We pick randomly two points, $p_1(x_1,y_1)$ and $p_2(x_2,y_2)$ inside a circle of origin $S$ with radius $R$ and we draw two circle $C_{1-2} (p_1,\sqrt {|x_1-x_2|²+|y_1-y_2|²})$ and $C_{2-1} ...
1
vote
1answer
252 views

What is the expected size of the convex hull of $n$-points selected randomly in a $2d$-circle?

We know $n>2$ and worst case its a triangle, best case the points all lie on a circle. Can we generalize to higher dimensions? What's the probability that the size of the convex hull of $n$ points ...
8
votes
1answer
165 views

A lawn, a flower, a pipe and the neighbours

You have a square lawn and a precious flower in the centre. You want to make sure you water the flower, and you don't particularly care how much of the lawn you water. To please your aleatory ...
9
votes
2answers
906 views

Expected area of the intersection of two circles

If we pick randomly two points inside a circle centred at $O$ with radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area ...
19
votes
5answers
6k views

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
2
votes
1answer
182 views

Expected area of the intersection of of triangles made up random points inside a circle, all the triangles must contain the origin

How to find the expected area of the intersection of a set of triangles made up $N$ random points that are picked uniformly inside a circle? The triangles must contain the origin of the circle. If ...
7
votes
1answer
173 views

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 ...
4
votes
1answer
187 views

What's the probability for two points to lie on the same side of the line joining two other points?

While trying to answer this question I realized that the probability for two points to lie on the same side of the line joining two other points is directly related to the probability for four points ...
13
votes
1answer
356 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
15
votes
1answer
303 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
10
votes
3answers
3k views

What is the probability that the center of the circle is contained within the triangle?

Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?
13
votes
2answers
470 views

Probability of circle given by randomly chosen diameter falling inside a square

Two dots are thrown into a square with side length 1 cm. The line ending in these two dots is the diameter of a circle. What is the probability that the circle lies in the square?
-5
votes
2answers
341 views

Geometry Probability Question

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Hi everyone I found this interesting question; help is ...
1
vote
1answer
81 views

Ring thrown in space

A ring is thrown randomly in the space and let $A(t)$ be its position at moment $t$. Let us say that the moment $t_{0}$ is "twisted", if the ring $A(t_{0})$ is linked with the rings $A(t)$ for $t$ ...
33
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
4
votes
2answers
617 views

probabilty of random points on perimeter containing center

related question: probablity of random pick up three points inside a regular triangle which form a triangle and contain the center What is the probability that a (possibly degenerate) triangle made ...
1
vote
3answers
1k views

Distance between $2$ random points in a segment.

On a straight line of length $10$ cm, two points A, B are selected at random uniformly and independently. What is the probability that the distance $AB > 4$ cm? Edit: I edited the question to make ...
4
votes
0answers
148 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
6
votes
0answers
177 views

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 ...
8
votes
2answers
411 views

What's the probability that three points determine an acute triangle?

Three distinct points are chosen at random from the unit square. The goal is to find the probability that they form an acute triangle. I started working on this because I want to know how to approach ...
1
vote
3answers
237 views

Cover a line segment randomly with smaller line segments

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
4
votes
3answers
193 views

Distribution of shapes of Delaunay triangles

Does anyone know the probability distribution of the shapes of Delaunay triangles in a constant-intensity Poisson process in the plane? Slightly later edit: One can imagine performing the experiment ...
18
votes
4answers
1k views

probablity of random pick up three points inside a regular triangle which form a triangle and contain the center

what is the probablity of random pick up three points inside a regular triangle which form a triangle and contain the center of the regualr triangle the three points are randomly picked within the ...
5
votes
1answer
351 views

Average area of choosing three points on a surface?

Assume I choose three random points on the surface of a sphere. What is the average area? (Each point is independently chosen relative to a uniform distribution on the sphere) Also, what would be the ...
3
votes
2answers
289 views

Probability on an infinite plane

(I thought this is a popular problem, but sadly Google yields nothing.) Three points are chosen at random on an infinite plane. What is the probability that they are on a line? And a variant: the ...