Probabilites of random geometric objects having certain properties (enclosing the origin, having an acute angle, being convex, ...); expected counts, areas, ... of random geometric objects.
1
vote
0answers
61 views
Spatial distribution of bees
* Please please help! I still get stuck.
We have a forest for bees, consisting of $4$ non-overlapping regions. $80\%$ of the bees seek honey in the forest while $20\%$ of the bees do so outside the ...
3
votes
1answer
95 views
Truchet tiles on a flattened cube
We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...
0
votes
1answer
14 views
craps game odd with a pair of dice
in a dice game craps, Alex rolls a pair of fair dice. if he gets 7 on the first roll, he wins immediately. if the result is any number other than 7, he keeps rolling the dice until he gets that number ...
1
vote
1answer
31 views
Probability of finding a point on or in an $n$-dimensional unit sphere
If a point is chosen at random in an $N$-dimensional unit sphere, what is the probability of falling inside the sphere of radius $0.99999999$? What if $N=3$, $N=10^{23}$, or $N = \infty $?
Okay, that ...
4
votes
2answers
53 views
finding the number of circles we get when randomly placing given patterns into a grid of squares
We have an 11$\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 ...
1
vote
1answer
34 views
Simple geometric distribution (solution verification)
The question is:
In a hockey competition, a player scores $80\%$ of his shots. What is the probability that the player will not miss until his $10^{th}$ try?
So I did the following
...
0
votes
0answers
20 views
finding the number of square we get when randomly put patterns into a given table [duplicate]
the image of the three tile patterns is here. [http://imageshack.us/photo/my-images/211/solpd.jpg/]
9
votes
1answer
209 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 ...
0
votes
2answers
32 views
Geometric probability question
So there are 2 parallel lines 20 feet apart. A piece of pipe 20 feet long falls between the lines and one end is exactly 10 feet from one line. What is the probability that the pipe lies entirely ...
1
vote
1answer
41 views
A modified Buffon's needle
A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it.
What is the probability that the needle lies between the two lines?
I can see how ...
1
vote
2answers
94 views
A person is selected at random from a population that has the following characteristics
A person is selected at random from a population that has the following characteristics:
...
2
votes
1answer
147 views
Expected area of the intersection of two and three circles
We pick randomly two points, $p_1(x_1,y_1)$ and $p_2(x_2,y_2)$ inside a circle of origin $S$ with radius $R$ and we draw two circle $C_{1-2} (p_1,\sqrt {|x_1-x_2|²+|y_1-y_2|²})$ and $C_{2-1} ...
1
vote
1answer
30 views
What is the expected size of the convex hull of $n$-points selected randomly in a $2d$-circle?
We know $n>2$ and worst case its a triangle, best case the points all lie on a circle. Can we generalize to higher dimensions? What's the probability that the size of the convex hull of $n$ points ...
4
votes
0answers
77 views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
4
votes
0answers
52 views
A lawn, a flower, a pipe and the neighbours
You have a square lawn and a precious flower in the centre. You want to make sure you water the flower, and you don't particularly care how much of the lawn you water. To please your aleatory ...
2
votes
1answer
100 views
Expected area of the intersection of of triangles made up random points inside a circle, all the triangles must contain the origin
How to find the expected area of the intersection of a set of triangles made up $N$ random points that are picked uniformly inside a circle? The triangles must contain the origin of the circle. If ...
8
votes
1answer
183 views
Expected area of the intersection of two circles
If we pick randomly two points inside a circle of radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the ...
8
votes
4answers
183 views
Probability that n points on a circle are in one semicircle
Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
2
votes
0answers
81 views
Sufficient Statistic for a Geometric R.V.
I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof.
The problem goes like this:
Suppose X is a discrete ...
1
vote
2answers
48 views
independent random variables geometric distributon
Suppose that $X$ and $Y$ are independent random variables with the same geometric distribution, $\mathbb{P}(X=k)=\mathbb{P}(Y=k)=pq^{k-1}$ for $k\geq 1$, $q=1-p$.
Find $\mathbb{P}(X=k \mid ...
1
vote
1answer
279 views
Proof of 2nd Derivative of a Sum of a Geometric Series
I am trying to prove how $$g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}$$
or $\sum ak(k-1)r^(k-1) = 2a(1-r)^{-3}$.
I don't know what I am doing ...
4
votes
1answer
61 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 ...
1
vote
1answer
68 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 ...
0
votes
1answer
38 views
Optimal $p$ for biased coin?
You are given a biased coin with probability $p$ of getting $H$ and $1-p$ of getting tail. Each flip is independent of another. We keep flipping the coin until we get $4$ consecutive tails. For each ...
7
votes
2answers
246 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 ...
10
votes
2answers
286 views
Circle and 2 dots
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?
5
votes
1answer
255 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 ...
15
votes
4answers
589 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 ...
14
votes
1answer
214 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 ...
6
votes
3answers
290 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?
8
votes
2answers
618 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?
3
votes
1answer
186 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 ...
-3
votes
2answers
270 views
Geometry Probability Question
Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Hi everyone I found this interesting question; help is ...
0
votes
0answers
60 views
Bernoulli Trials with two random variables - Run of successes and faliures
Consider a sequence of Bernoulli trials with probability of success $p$. Suppose you started the game with a run of succsses followed by the run of faliures (note that you can learn an unlucky run is ...
1
vote
1answer
66 views
ring thrown in space
A ring is thrown randomly in the space and let $A(t)$ be its position at
moment $t$. Let us say that the moment $t_{0}$ is "twisted", if the ring
$A(t_{0})$ is linked with the rings $A(t)$ for $t$ ...
0
votes
2answers
165 views
How can I calculate the CDF of this random variable?
$X_1$, $X_2$, $X_3$ are random variables distributed following non-identically independent exponential distribution.
The PDF $X_i$, $f_{X_i}(x)$=$\frac{1}{\Omega_i}\exp(\frac{x}{\Omega_i}), ...
0
votes
2answers
217 views
Binomial/Geometric Distribution explanation
I've found the following exercise in my Stats coursework. I only have solutions to it, but no explanation. And I would really like to know how to get to the answer.
An urn holds 5 white and 3 black ...
1
vote
0answers
80 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 ...
2
votes
0answers
87 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 ...
1
vote
0answers
69 views
geometric sum with probabilities
I have the following sum to consider:
$$
p_i/(1+r)^i
$$
for i = 1 to 10
My question is, whether this can be somehow evaluated using some sort of mathematical insight without explicitly computing ...
2
votes
2answers
145 views
What is the appropriate probability distribution to model this situation?
I want to model a random variable which represents the number of failures before success in a repeated Bernoulli trial. I will conduct only utmost N trails and I am ...
1
vote
3answers
133 views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
2
votes
2answers
127 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 ...
