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.

learn more… | top users | synonyms

1
vote
1answer
30 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
0
votes
0answers
50 views

What is the extent of the streaks covering a square? [closed]

Let $N$ be a unit square, $1 < A <\sqrt{2}$ real number, we put a strip with width $A$ to the square randomly. I would like to determine the measure of the strips, that cover the square. I don't ...
1
vote
1answer
50 views

Concluding from limiting behavior

I've recently seen the following question on the internet: If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around? ...
0
votes
0answers
23 views

What is the probability of uniformly sampling a point in d-dimensional hypercube?

Let us consider a hyper-cube whose length is l units along each of its d-dimensional structure. It is desired to uniformly sample a point inside the hyper-cube. How to do uniform sampling and what ...
0
votes
1answer
47 views

Expected area of an internal triangle determined by a random point in a triangle

A point M is chosen at random (uniformly) inside an equilateral triangle ABC of area 1. How to find the expected area of the triangle ABM?
-1
votes
1answer
38 views

Expected nof children “at least one boy and at least one girl, with boy older than girl”

A couple decides to keep having children until Cond1: they have at least one boy and at least one girl, Cond2: with boy older than girl and then stop. Assume they never have twins, that the ...
0
votes
0answers
38 views

How long does one remain in the square

This is a slightly known question: There is a square of unit side. Its centre is $O$. Two movable points $X$ and $Y$ are placed randomly in the square. Let $A$ be the midpoint of $OX$ and $B$ ...
0
votes
1answer
14 views

Issue with sum of probabilities of probability distribution function of a geometric random variable

Is it possible that the sum of probabilities of geometric distribution for "$k = 1,...,n$", where k is number of trials until the first success, is not equal to 1? I'm asking this, because I encounter ...
2
votes
1answer
102 views

Three points on sides of equilateral triangle

Let's choose three points on the sides of an equilateral triangle(one point on each side) and construct a triangle with these three points. what is the probability that area of this triangle be at ...
0
votes
1answer
9 views

Proof that 2 geometric random variable is NB

can someone write me the proof of 2 geometric variable are negative binomial ? $X\sim G(p)$ and $Y\sim G(p)$ how can i proof that $Z=X+Y \sim NB(2,p)$?
8
votes
1answer
46 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 $AC$ ...
5
votes
1answer
24 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. ...
0
votes
2answers
55 views

Probability of segment lying in circle

Given a circle of radius R: $x^2+y^2\le R$, find probability of horizontal segment with length $\frac{R}{2}$ lie whole inside this circle. Position of segment's center has uniform distribution in ...
0
votes
0answers
8 views

Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
1
vote
0answers
29 views

Generate random numbers from appropriate distributions

Generate random numbers from appropriate distributions to find the area of the region enclosed by the curves y = sin (cos(x)), y = 0, x = pi/2 , and x = -pi/2 and report the area.
0
votes
1answer
37 views

Probability of receiving a correct packet of N bits

When a packet is transmitted on a communications link, the probability that a bit in packet is received in error is p. Assume that the packet has N bits. Suppose the packet length is random i.e. N is ...
3
votes
1answer
181 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 ...
1
vote
0answers
76 views

Positivity of density for sum of dependent random variables

Let $\{\xi_i\}_{i\geq 0}$ be a sequence of iid random variables that are uniform on a d-dimensional box $B_1(0) = [-1,1]^d$. Let $\{A_i\}:\mathbb{R}^d \to \mathbb{R}^d$ be invertible matrices with ...
0
votes
0answers
12 views

Is there a way of solving this probability question without using the survival characteristic of a geometric random variable?

I have the following problem, and the author presents the solution by using the survival characteristic of a geometric random variable: However, I am not very familiar with this method and was ...
1
vote
2answers
29 views

Sum of geometric and Poisson distribution

Suppose I have $X \sim \mathrm{Geom}(p)$ and $Y=\mathrm{Pois}(\lambda)$. I want to create $Z = X + Y$. (Note: the $X$ begins at $0$ rather than $1$.) Is this possible, please? Then I would calculate ...
0
votes
0answers
19 views

How is the Dehn invariant related to the mean width?

Reading Ravi Vakils Monthly article of february 2011 and watching the video; he mentions that the Dehn invariant is related to the linear invariant measure $\mu_1$ of geometric probability. The Klain ...
9
votes
2answers
901 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 ...
0
votes
0answers
106 views

The Curvy Rebound: One of the most interesting (Geometric) Probability problems.

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also ...
9
votes
5answers
372 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: ...
0
votes
1answer
34 views

Geometric probabilities with rectangle

One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than ...
1
vote
0answers
34 views

Geometric probability with square

Jhon and Simon have common bank account which has $720$ dollars. Each of them has to buy a gift independently from other (Gift cost $< 360$ dol). What is the probability that after shopping there ...
0
votes
0answers
22 views

The expected number of polygons created as a result of the intersection between randomly placed rectangles inside a square

How can I compute the expected number of polygons created as a result of the intersection of $k$ rectangles of area $B$ each, which fall randomly inside a square of area A? Regarding how randomness ...
2
votes
1answer
102 views

Probability of intersection of line segments

A pair of points is selected at random inside a unit circle and a line segment is drawn joining the two. Another pair is selected and a second line segment is drawn. Find probability that the two ...
11
votes
1answer
221 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 ...
3
votes
1answer
73 views

Expected length of minimum chord

You are given a circle of radius $1$. Suppose you pick $n$ independent points randomly on the circle and join neighboring points with lines to create chords. What is the expected length of the ...
33
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 ...
6
votes
1answer
115 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 ...
0
votes
1answer
55 views

PDF of distance from the center of a random point in the unit disk

I found in a certain website (also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: $P(d)=2d$, but no one is giving the way to derive ...
2
votes
1answer
76 views

Probability of average distance from origin of unit circle less than half

Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than half?
0
votes
2answers
146 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 ...
13
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?
9
votes
4answers
223 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
131 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 ...
11
votes
4answers
3k views

What is average distance from square center to some point?

How can I calculate average distance from square center to points inside the square?
2
votes
1answer
26 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
votes
2answers
67 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 $k$?
4
votes
1answer
121 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
vote
0answers
23 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
vote
0answers
55 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
votes
2answers
44 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, ...
3
votes
3answers
125 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 ...
13
votes
1answer
356 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 ...
11
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 ...
2
votes
1answer
102 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 ...
3
votes
1answer
113 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 ...