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 a coin falling on the edges of a square

Let a coin be randomly (and uniformly) dropped onto a square on the floor. Assume the edge length of the square to be $ d $ and the radius of the coin to be $ r < d/4$. I know that the probability ...
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2answers
49 views

2 pairs of points on a circle.

Two pairs of points are randomly chosen on a circle. Find the probability that the line joining the two points in one pair intersects that in the other pair. I've been thinking over this problem, ...
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3answers
132 views

Expected area of a random triangle with fixed perimeter

I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle ...
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1answer
137 views

Expected time to completely cover a square with randomly placed smaller squares

Suppose I have the unit square $[0,1]^2$ and I choose a point $(x_1, y_1)$ randomly in a uniform manner inside $[0,1]^2$ and draw a filled in square of side length $1/N$ with center $(x_1, y_1)$. And ...
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1answer
44 views

Bertrand paradox Random midpoint

http://en.wikipedia.org/wiki/Bertrand_paradox_(probability) The link above explains Bertrand paradox in probability. In "Random Midpoint method" Bertrand uses a concept that all chords whose ...
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1answer
75 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
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1answer
126 views

Circular distribution of circles

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of ...
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1answer
77 views

expected size of a special set of random points in the unit square

Today I came up with this fun problem, but I'm having a hard time to solve it completely myself. The question is the following: Let's generate n random points ...
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1answer
53 views

probability for two vectors to lie on different regions created by hypeplane

Suppose we have two vectors $v_i,v_j$ and there is one hyperplane whose normal is chosen uniformly from the unit sphere. Then what will be the probability that $v_i$ lies on one side and $v_j$ lies on ...
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1answer
54 views

fixed length random chord outside of circle.

consider a uniform distribution on a unit circle, I construct a cord by the following steps: pick one endpoint A within the unit circle uniformly. points that are $0<d<1$ distance away from ...
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1answer
76 views

Probability that a triangle has an angle greater than 120 degrees

We've got a circle and we draw $3$ points, which form a triangle. Question: what is the probability that its greatest angle has more than $120$ degrees? Well, I have no idea how to do it. I know some ...
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41 views

Shortest path length when edge length is limited

$N$ nodes are uniformly distributed in a square whose side length is $1$. There exists an undirected edge between two nodes, if and only if the distance between them is less than or equal to $r$. Here ...
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1answer
36 views

Density on the square, expected value

Let $f: [0,1]^2 \rightarrow \Bbb R^{+}$ a density function on the square. I suppose that the random variable $X=(X_1,X_2)$ has the density f with respect to the lebesgue measure. I denote ...
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1answer
85 views

Random embedding of $K_4$ in the unit square

Suppose I embed $K_4$ (the complete graph on 4 vertices) randomly in the unit square (using the uniform distribution for the positioning of the vertices). $K_4$ is planar, but not any embedding of it ...
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1answer
164 views

A point is selected uniformly at random in the interior of a unit square…

From It, the altitude to each side of the square is drawn. For each side, a stick of the altitude's length is obtained. Determine the probability that you can select three of the sticks and arrange ...
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3answers
524 views

find a chance that all N points lie on the half circle. [duplicate]

We are given a circle with N randomly allocated points on it. Task is to find a chance that all N points lie on the one half of circle. I have drafted some solution: 1. Since there are no way to put ...
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1answer
984 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 $b$. ...
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1answer
134 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 ...
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1answer
164 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 ...
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2answers
163 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 ...
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1answer
120 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 ...
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2answers
613 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 ...
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1answer
172 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 ...
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1answer
197 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 ...
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1answer
177 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 ...
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1answer
178 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 ...
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84 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 $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
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1answer
86 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 ...
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86 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 ...
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0answers
96 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 ...
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1answer
988 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 ...
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0answers
47 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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2answers
450 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 ...
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1answer
262 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 ...
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1answer
248 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 ...
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2answers
112 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 ...
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1answer
185 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 ...
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2answers
208 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 ...
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1answer
93 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 ...
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2answers
110 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 ...
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1answer
797 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} ...
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1answer
302 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 ...
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1answer
183 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 ...
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2answers
1k 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 ...
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5answers
7k 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.
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1answer
186 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 ...
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1answer
194 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 ...
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1answer
221 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 ...
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1answer
373 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 ...
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1answer
317 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 ...