Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

-1
votes
1answer
36 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 ...
0
votes
0answers
18 views

Is the following approach to this simple probability problem correct?

Problem: 1000 bacteria are randomly distributed over 100 fields (all the same size) next to each other. Wanted: Expectation for how many bacteria there are per field The teacher's way is: ...
1
vote
0answers
22 views

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

My question sounds like this: In the square K = {(u,v) : u,v ∈ [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 ...
0
votes
1answer
27 views

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

my question sounds like this: 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 ...
0
votes
2answers
46 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 $ ...
1
vote
0answers
30 views

The possibility of cutting 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 ...
0
votes
1answer
31 views

Probability of two buyers service time

I have a geometric probability question and I don't know how exactly it should be solved and how graph should look like. The question is: Serving one person takes from $8$ to $22$ minutes time. ...
0
votes
0answers
33 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-\...
1
vote
1answer
37 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 ...
0
votes
1answer
56 views

The sum of the total number of service representatives who spoke with all the customers who called during a minute.

I trying to solve this problem, but I can not tell if what I am doing is right, At some company, the customer support department gets called by $5$ customers per minute independently. The ...
0
votes
2answers
36 views

Net gain on dice rolling

Suppose you have to pay $15 to roll a dice repeatedly until the first 6 comes up. Then you will get paid as many dollars as the square of the no of rolls (X^2). I have a equation as follows: G = Gain ...
1
vote
0answers
17 views

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 ...
0
votes
0answers
31 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 ...
0
votes
2answers
63 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 ...
1
vote
1answer
60 views

Poisson Variable with an Exponential Parameter becoming a Geometric Distribution?

Suppose Λ ∼ exponential(γ) and X ∼ Poisson(Λ). Use moment generating functions to show that X + 1 ∼ geometric(p) and determine p in terms of γ. In order to solve this problem, I first did: $E[e^{s(X+...
0
votes
0answers
25 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,...
0
votes
1answer
27 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 ...
0
votes
1answer
96 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 .
1
vote
0answers
29 views

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 ...
2
votes
1answer
131 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). ...
0
votes
1answer
56 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 ...
1
vote
0answers
59 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 ...
6
votes
0answers
76 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 ...
1
vote
2answers
53 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 ...
0
votes
1answer
43 views

Expectation problem with geometric random variable?

If I have a stream of integers from 0 to 9(each with equal probability and repetition is allowed) that are spewed out until I get the first instance of 4-5 next to each other in this order, how do I ...
0
votes
1answer
29 views

$\min\{X, Y\}$ is geometrically distributed according to parameter $1 - (1-p)^{2}$

Let $X,Y$ be independent from the parameter $0 < p < 1$ distributed random variable on a discrete probability space $(\Omega, \mathcal{F}, \mathbb P)$. Show that $\min\{X,Y\}$ is geometrically ...
11
votes
1answer
183 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 ...
1
vote
0answers
68 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 $A_{n}$. ...
2
votes
0answers
35 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}(|...
0
votes
0answers
34 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 ...
4
votes
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 ...
1
vote
1answer
132 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 ...
6
votes
3answers
177 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?
0
votes
1answer
145 views

CDF and Survival Function of Geometric Distribution

I am having trouble understanding the intuition behind the CDF and survival probability of a geometric distribution on both {0, 1, ...} and on {1, 2, 3, ...}. I know that a geometric starting from 0 ...
0
votes
2answers
22 views

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 ...
0
votes
1answer
22 views

Probability of joint distribution has closed form answer?

Let $Y$ be a geometric distribution with a parameter $\theta$ such that $0 < \theta < 1$ and let also $P(X=x|Y=y)$ be a binomial distribution with parameters "y" and "1/2". So: $$ P(X=x|Y=y) = ...
0
votes
2answers
47 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 $...
1
vote
0answers
40 views

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 $\...
1
vote
0answers
41 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 ...
0
votes
0answers
32 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\...
4
votes
1answer
53 views

Geometric probability - challenging problem(two points of a square K determine a diagonal of another square that is contained in given square 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 $AB$, ...
3
votes
1answer
37 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 ...
0
votes
1answer
101 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 ...
2
votes
0answers
31 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{\...
1
vote
1answer
27 views

Geometric distribution random

In reading a research paper I came across this note, I am not sure I fully understand it. I am wondering if they are referring to P as the geometric distribution so their output would be $p^r$ where $...
0
votes
0answers
35 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 $t=t_0$ ...
1
vote
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 ...
0
votes
1answer
102 views

Probability Question Help- Statistics, Distribution, Geometric Distribution and Normalization.

I am having trouble with a probability question regarding to G-distribution. Background: Let's assume I have a disc that has two sides, one is black and one is white. Note: the disc has biased ...
2
votes
2answers
194 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 ...
3
votes
1answer
201 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 ...