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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.

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Random Variable Y following uniform distribution with parameter Random X that follows geometric.

Random variable X follows geometrical distribution with p=1/4. Random variable Y follows uniform distribution in [-X,X]. I'm looking for P(Y>3/2) and also P(X=2|Y>3/2).I know for a fact that Σ(from k=...
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14 views

Geometric Probability. Sphere

Suppose a sphere with radius $R$. Find the probability of the event such that $n$ selected points of the sphere are within the distance of $r = \frac{R}{2}$ from the center of the sphere. The points ...
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1answer
59 views

Average area of the shadow of a convex shape [closed]

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 ...
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1answer
25 views

How to sample uniformly from the surface of a (fish-) bowl?

Define a fish-bowl as a sphere comprised between two horizontal disks. That is, a sphere where we have replaced the top and bottom sectors by horizontal disks. See picture below. How to sample ...
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25 views

What is the probability that the angle is a right angle?

Let's say you have a square, say $ABCD$, with side length 1. A point $P$ is randomly chosen from inside the square, with any point being equally probable of being chosen. What is the probability that ...
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1answer
23 views

Expected value of number of dots in random placed circle

Given a square with vertices at $(0,0), (1,0), (0,1), (1,1)$ and $N$ labeled dots in this square with coordinates $(x_i, y_i)$. Bowser picks a dot $(a, b)$ in square (with uniform distribution) and ...
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2answers
63 views

Special case of Bertrand Paradox or just a mistake?

I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
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1answer
38 views

Randomly choose $n+1$ points on a $S^{n-1}$, probability of $n$-simplex containing center

Randomly choose $n+1$ points on a $S^{n-1}$(surface of ball in $n$-dim space). What's the probability that the $n$-simplex formed by these $n+1$ points contain the center of the sphere? I conjecture ...
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60 views

Method to build a polyhedral die with given probabilities

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
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2answers
59 views

What is wrong with my method for probability that n points on a circle are in one semicircle

So I understand the method used in this solution, and I know my method is incorrect, but I was just looking for an explanation why. I was thinking that if I choose any spot on the circumference, ...
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1answer
50 views

Which cell in a grid a point belongs to?

I have a square which is divided into an NxN grid and I have a dot. How I can find which cell this point belongs to? Assuming I know the boundaries of the square. How to find the point belongs to the ...
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44 views

What's the probability that equation $x^2 + px + 6.9q = 0$ has real solutions

In the square $K = \{(u,v): u,v \in [0,\;9.6]\}$ randomly selected point with coordinates $(p,q)$. What is probability that equation $x^2 + px + 6.9q = 0$ has real solutions? I was trying to solve ...
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1answer
35 views

What's the probability that triangle area will be smaller than 7.25

Rectangle edges equals a=2.9 and b=6.3. In adjacent rectangle edges randomly selected two points and straight line drawn through them. What is the probability that drawn triangle area is smaller ...
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2answers
51 views

What's the possibility of inequality

It is known that $x\in [0,10]$. What is the possibility that $x^{2}+b > ax$ is true, when $a = 15.3$ and $b = 58.5$. Is it correct to calculate this like that: $x^{2} + b$ is more than $ax$ when $ ...
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40 views

The probability of cutting data (recording) on a tape

In an archive there was a 200 meter long magnetic tape. On one side of the tape there was a recording that takes up 54 meters of the tape and on the other side, there was a recording that takes up 13 ...
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46 views

When is the measure of spherical cap large?

It is known that in high dimensions, the measure of the spherical cap is small, due to the measure concentration for the sphere. In particular, we have the following inequality in $n$ dimension: $$ 1-\...
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1answer
48 views

What is the probability if we throw dart towards a large square but it should hit only the inner part of small square $FEHG$ inscribed in it?

Let $ABCD$ be a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are $x$ and $y$, where $\frac{x}{y}$ = $3$. A dart is thrown towards the square ...
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Brownian-like motion covering a set of positive measure

I've been wondering about this question, in $\mathbb{R}^2$ under what conditions does a Brownian-like motion $BL_t$ covers a set of positive measure or, what are the odds that, for some initial ...
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34 views

Probability of two cars crashing exiting a car park

A car is stationary at the center of a car park which has an area C. A blind man is driving and sets off to exit the car park in a random direction at a known constant velocity Va. A second car in the ...
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2answers
65 views

Average of shortest half of two halves and probability of three halves making a triangle

This is a problem that has been bothering me because it seems so easy however the answers don't feel right so...(It showed up in my latest statistics exam and almost everybody got it wrong because we ...
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0answers
29 views

Geometric probability independence

I have a basic question about the independence of two events. Let $x$ be some fixed point in the interior of some set (say it's a convex bounded set $C$ in $\mathbb{R}^2$ or something). Choose $a, b,...
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1answer
31 views

Minimum number of random vectors needed to span a space

I am working with $\mathbb{R}^{nm}$ for some $n,m\geq3$. What is the minimum number of random vectors I need for them to span $\mathbb{R}^{nm}$ (in terms of $n$ and $m$)? I'm happy for this to ...
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1answer
105 views

Expected area of triangle inscribed in a circle

On a unit circle , BC is a chord with length 4/π . Point A is picked randomly at its circumference . Find the expected area of △ ABC .
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generalization of Dyck Path: size K upward steps

One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 ...
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1answer
179 views

Probability of two random points being orthogonal in higher-dimensional unit sphere

I understand that most points will be close to surface due to volume concentration. Also I also understand the concentration of volume near the equator, relative to any specific point (North pole). ...
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1answer
75 views

Expected distance between two points on randomly selected line segment

My question is similar to a question answered before many times, namely the expected distance of L/3 between two random points on a line segment (Average Distance Between Random Points on a Line ...
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0answers
62 views

Answer To A Probability Problem About Placing Random Points In A Circle.

Problem Statement: Three dots are randomly placed in a circle of radius one cm. What is the probability that when a fourth dot is placed (randomly) in the circle, it is at least one cm away any of the ...
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0answers
81 views

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 ...
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2answers
57 views

Expected value of the number of steps in a walk around the rectangle.

We have a rectangle $ABCD$. The walk starts at $A$. Let $X$ be the time required to visit all the vertexes. (We independently go to the adjacent vertex with probability 1/2). Calculate $EX$. Some ...
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1answer
192 views

Voronoi cell volume inside the ball

I have the following problem: Let us denote a ball with center $C$ and radius $R$ in $\mathbb{R}^d$ as $B(C, R)$. Given a unit ball $B(0, 1)$ and vector $u$ has a uniform distribution inside the ...
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71 views

Find probability that a line l may be tangent to circle $x^2+y^2=n^2\left(1-(1-\frac{1}{\sqrt n})^2\right)$

Consider the set $A_{n}$ of points $(x,y)$ where $0\leq x\leq n,0\leq y\leq n$ where $x,y,n$ are integers. Let $S_{n}$ be the set of all lines passing through at least two distinct points of $A_{n}$. ...
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38 views

Expected minimum absolute value of unifrom distribution

Consider we draw N iid values $x_1, x_2,...,x_N$ from an uniform distribution on $[- \epsilon, \epsilon]$. Can we calculate the expected minimum of their absolute values: $$ \mathrm{E}(\mathrm{min}(|...
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40 views

Nearest distance distribution from a node

Please see the scenario below. I am interested in finding the probability density function of green point to its nearest blue point in circle with Radius Rc. This must not include any point within ...
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1answer
70 views

Find the probability that A,B,C are connected

I was given the following problem as a homework assignment: Denote with $S$ the ball with center $O$. Three points $A, B$ and $C$ are chosen at random on its surface, their positions being ...
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1answer
160 views

Expected length of arc containing a point

You are given a circle $C$ of unit radius and a point $M$ on it. Now, three points are chosen at random on the circle $C$ which divides it into three arcs. What is the expected length of the part that ...
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3answers
224 views

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?
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2answers
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What is the probability that coin will not touch lines of segments?

In geometric probability I want to know what is the probability that a small coin (r < a) will not touch the lines of segments with 2*a distance(I mean the plain is divided into 2*a length segments ...
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2answers
49 views

A geometric probability question

Find the probability of distance of two points ,which are selected in $[0,a]$ closed interval, is less than $ka$ $k \lt 1$ What did I write : $P(A)$ = (Area measure of set $A$)/(Area measure of set $...
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What is the surface area an ellipse takes up on a sphere?

Take $\Sigma$ a $k\times k$ positive-definite real matrix and $E$ to be an associated ellipse: $$E:=\{(x_1,\dots, x_N): \frac{1}{N}\sum_n x_n^\dagger \Sigma x_n \leq 1\}.$$ Now take $z$ uniform on $\...
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0answers
43 views

PDF Modification from a uniform to a way-point spatial distribution

Basically, I know that for a uniformly distributed points, r within the intersection $XQYD$ in Figure 1, the pdf is given by $$f(r) = \frac{l(r)}{A}$$ where A is the area of the intersection region ...
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42 views

Conditioning a PDF in the intersection of two circles.

Given the diagram below (The first figure): $$f_r(r)= \frac{\lambda \pi r \exp{(-\lambda \pi r^2/2)}}{1-e^{-\lambda \pi R^2/2}}$$ where $$0<r\leq R$$ The pdf $$f_{\text{Z}}(\text{z})=\frac{2\...
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1answer
69 views

Two points of a square $K$ determine a diagonal of another square that is contained in $K$

Let $K:=[0,1]^2$ be a square on $\mathbb{R}^{2}$. We select 2 random points $A$, $B$ $\in [0,1]^{2}$ in this square. What is the probability that the square whose diagonal is the line segment $AB$, is ...
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1answer
38 views

What is the probability that 3 random points in $2n$ regular polygon contains the center of gravity

What is probability that $3$ random point in $2n$ reguler polygon contains center of gravity I contain the diagram to make up my explanation. Left one is contain the center of gravity,while right ...
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1answer
140 views

Probability that the distance between two random points inside a circle is less than some value

Two points are chosen at random within a circle of radius R. What is the probability that the distance between them is less than D? I can solve the problem for the special case $R=D$ as follows: For ...
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34 views

Bounding probability of picking point from $l^p$ ball [closed]

Consider picking $x$ randomly and uniformly from the $l^p$ ball with radius $R$, $B_p^d(R)$. If $0\le \gamma \lt 1$, how can I conclude following? $$ P(\| x\|_p \le \gamma R) = \frac{\...
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0answers
37 views

Probability of finding nodes in a moving circle

This seems to be a simple problem to me, but am not getting right results. As attached in the figure (not drawn to scale). A circle centered at $X$ (X in Blue) with initial position at time $t=t_0$ ...
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1answer
52 views

Difference of numbers in a unit interval

$x,y,z\in \mathbb{R}$ are chosen at random from the unit interval $[0, 1]$. What is the probability that $$\max(x,y,z) - \min(x,y,z) \leq \frac{2}{3}$$ EDIT- Solutions not using calculus would be ...
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2answers
247 views

Interview Question on Probability: A and B toss a dice with 1 to n faces in an alternative way

A and B toss a dice with 1 to n faces in an alternative way, the game is over when a face shows up with point less than the previous toss and that person loses. What is the probability of the first ...
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1answer
217 views

Average distance detween two random points on two line segments

Suppose you have two straight line of length $L_1$ and $L_2$, and a point is chosen at random along each line. What is the expected distance between these points? This question is a complement of ...
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Probability that a random graph will remain planar after adding an edge

According to this answer, a random graph on $n$ vertices is a graph which has each of the $n\choose2$ edges independently with probability $1/2$ each. The probability of at most $3n-6$ edges (which is ...