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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76
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4answers
6k views

The “pepperoni pizza problem”

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...
43
votes
3answers
1k views

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
34
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 ...
19
votes
5answers
10k 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.
18
votes
4answers
2k 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 ...
16
votes
1answer
332 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 ...
14
votes
2answers
2k views

Probability that the convex hull of random points contains sphere's center

What is the probability that the convex hull of n+2 random points on n-dimensional sphere contains sphere's center?
14
votes
1answer
396 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 ...
13
votes
2answers
538 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?
13
votes
1answer
198 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 ...
12
votes
4answers
5k 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?
12
votes
3answers
2k views

Simulating uniformly on $S^1=\{x \in \mathbb{R}^n \mid \|x\|_1=1\}$

A scheme to generate random variates distributed uniformly in $S^2=\{x\in \mathbb{R}^n \mid \|x\|_2=1\}$ is well known: generate a standard normal variate in $\mathbb{R}^n$ and normalize it to unit ...
12
votes
1answer
241 views

Expectancy value for the percentage of points lying in the Convex Hull (3D)

Suppose I chose n uniformly distributed random points in a 3D cube. What is the expected value for the percentage of points lying on the convex hull as a function of n? Just as a reference, I made ...
11
votes
4answers
4k views

What is average distance from center of square to some point?

How can I calculate average distance from center of a square to points inside the square?
10
votes
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 ...
9
votes
5answers
558 views

Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary

Background : I happen to love solving tough problems. Problem is, I simply cannot answer some! It happened again today, as I attempted to solve the questions in this site: ...
9
votes
4answers
317 views

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is uniform) independently of others We construct new triangle B from ...
8
votes
2answers
470 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 ...
8
votes
1answer
177 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 ...
8
votes
1answer
165 views

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 $AC$ ...
8
votes
1answer
193 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 ...
7
votes
3answers
269 views

Probability that one of a set of four points lies within the triangle formed by the other three

Given four points, each randomly chosen with a uniform probability distribution in the interior of a (WLOG unit) circle, what is the probability that (any) one of the points lies within the triangle ...
7
votes
1answer
215 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 ...
7
votes
1answer
162 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 ...
6
votes
2answers
368 views

What is the probability that a quadrilateral is convex?

Given $4$ distinct randomly chosen points $x_1$, $x_2$, $x_3$, and $x_4$ in the plane such that the polygonal path from $x_1$ to $x_2$ to $x_3$ to $x_4$ to $x_1$ describes a non-self-intersecting ...
6
votes
0answers
206 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 ...
5
votes
1answer
388 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 ...
5
votes
2answers
796 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 ...
5
votes
1answer
82 views

Show the following pdf is memoryless

I've been thinking about this around 2 weeks for the midterm. but still can't prove it. I used this $$ P(X > r+s | X > s) = P(X > r) = \mathrm e^{−\lambda r}$$ $$P(X > r + s) / P(x ...
5
votes
3answers
624 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 ...
5
votes
2answers
113 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 ...
5
votes
1answer
39 views

PMF for K, the number of trails up to, but not including, the second success

I'm taking an MIT OCW course on Probability. Question: Al performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of three fair coins. ...
4
votes
2answers
1k views

Random walk on a cube

Start a random walk on a vertex of a cube, with equal probability going along the three edges that you can see (to another vertex). what is the expected number of steps to reach the opposite vertex ...
4
votes
1answer
1k 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
3answers
203 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 ...
4
votes
3answers
186 views

Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...
4
votes
1answer
168 views

Selecting a Random Point Inside a Cube

A point $P$ is selected at random inside a cube. Find the probability that $\angle APB \geq 135^o$, where $\overline{AB}$ is a body diagonal of the cube. I am not able to come up with the right ...
4
votes
1answer
266 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 ...
4
votes
2answers
789 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 ...
4
votes
1answer
94 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 ...
4
votes
1answer
143 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 ...
4
votes
0answers
163 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 ...
3
votes
4answers
644 views

Random walking and the expected value

I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me. Lets say we have a square, with vertex's 1234. I can randomly walk to each ...
3
votes
2answers
340 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 ...
3
votes
1answer
196 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 ...
3
votes
3answers
146 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 ...
3
votes
1answer
97 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 ...
3
votes
1answer
93 views

The expected value of the $k$th order statistic of iid geometrically distributed rvs, and its asympotic expansion.

I have read the paper Combinatorics of geometrically distributed random variables: Left-to-right maxima. In the paper, the largest order statistic $X_{n:n}$ (i.e., $\max\{X_1,X_2,\ldots,X_n\}$) is ...
3
votes
1answer
137 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 ...
3
votes
1answer
101 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 ...