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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2answers
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What is the average number of probes required to insert a new record into the hash table for this student? [on hold]

Suppose a college has a master file consisting of $1000$ student records. Suppose that a hash table has been constructed to hold $3500$ records, and currently holds $1000$ records. A new student ...
8
votes
4answers
163 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 ...
7
votes
3answers
103 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 ...
2
votes
1answer
22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
0
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2answers
38 views

Probability of two points being a certain distance apart on a circle

Is the probability of two points being a certain distance $k$ apart on a circle of length $m$, with $0\le k<\frac{1}{2}m $, always the same for any $k$?
1
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1answer
34 views

Progressive Roulette Betting

In this problem we discuss a betting strategy known as "progressive betting". Here's the setup: Bets are repeatedly made at a roulette according to the following strategy: All bets are made the ...
3
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3answers
30 views

Geometric distribution with multiple successes

Here's the question: "A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is $u$, let $X$ = the number of doors he knocks on. Find ...
4
votes
1answer
90 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 ...
1
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0answers
21 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
1
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0answers
38 views

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 ...
0
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2answers
39 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, ...
1
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2answers
151 views

Grid problem - combinatorics? [closed]

In a city, streets are laid out as a grid. Your home is at location (0,0) and work is at location (10,10). You walk to work taking only steps up and to the right. (a) How many distinct ways can you ...
3
votes
3answers
112 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 ...
2
votes
1answer
67 views

Expectation of Geometic distribution

I have the following question : It costs $30$ cents per day to keep pigeons. Let $N$ be the number of pigeons kept and suppose that $N$ has the geometric distribution $Pr(N=n) = \frac1{10} ...
0
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0answers
59 views

Find probability of angle being obtuse

We are given points A and B on the 2D plane and distance between them is 2. Let C - randomly picked point on the circle with radius R and center at the middle of AB. Find probability of angle ABC ...
0
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1answer
26 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 ...
2
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1answer
59 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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0answers
36 views

Radon transform, Buffon's needle and Integral geometry

In all the literature that I have seen it is mentioned that these two are "branches" of integral geometry, but no where I can see the exact connection since one is connected with probability and the ...
3
votes
1answer
100 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 ...
2
votes
1answer
52 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 ...
0
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1answer
44 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 ...
0
votes
1answer
48 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 ...
3
votes
1answer
63 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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0answers
23 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 ...
0
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1answer
32 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 ...
4
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1answer
75 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 ...
1
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1answer
102 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 ...
5
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3answers
344 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 ...
3
votes
1answer
751 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
102 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 ...
2
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0answers
28 views

Geometric Probability Question Without Calculus [closed]

Three people go to the same place the same day. Each shows up at a random time within a 12 hour span. The first remains at the place for 2 hours, the second remains for 1 hour, and the third remains ...
7
votes
1answer
140 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
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0answers
13 views

Probability of error in two dimensional signal space

The likelihood decision rule for two dimensional signal space is $r_2$ > $r_1$ for $H_1$ hypothesis and $r_2$ < $r_1$ for $H_0$ hypothesis. The range of $R=[r_1~r_2]$ is $-\infty<r_1<\infty$ ...
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0answers
36 views

Boundary points of probability simplex

I have a very simple question for which I know the answer but I can not prove it! What are the boundary points of a probability simplex? I know every probability vector with one zero component lies ...
0
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1answer
218 views

Project Euler #453 confusion

So I decided to give a shot on the #453 project euler problem but there is something that confuses me with the numbers given. I decided to start by calculating the possible arrangements of 4 vertices ...
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2answers
107 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 ...
1
vote
1answer
74 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
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2answers
377 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
117 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
152 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
122 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
130 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
75 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
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1answer
73 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
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0answers
73 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
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0answers
65 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
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1answer
396 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
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0answers
41 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
342 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 ...