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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Sufficient condition for 3 positive scales to represent sides of a triangle [duplicate]
Now with the above condition check the following video at 4:00
Check https://youtu.be/Zlnpo3GxWpY?si=Pagh6pcxgmRmEUHy
At 4:00
Any one of the inequalities (written with red color) should be sufficient ...
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If $(a,b,c)$ are the sides of a triangle, what is the probability that $ac>b^2$?
Let $a \le b \le c$ be the sides of a triangle inscribed inside a fixed circle such that the vertices of the triangle are distributed uniformly on the circumference.
Question 1: Is it true that the ...
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Probability for the highest correlation between random vectors
Let $k_1,\dots,k_m\in\mathbb{R}^d$, denote by $\mathcal{D}:=\mathcal{N}\left(0,\frac{1}{d}I_d\right)$ and $[n]= \{ 1,\dots,n \} $. I am interested in the following probability:
$P_{ x_1,\dots,x_n\sim \...
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Buffon's needle in one dimension
I want to solve Buffon's Needle problem but first I was trying to tackle a simpler case.
So: consider an infinite line with points each $t$ units. Let's say that we have a "needle" of length ...
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Probability that two random lines intersect inside a square
Consider the square with vertices $(0,0),(1,0),(1,1),(0,1)$.
Choose two independent uniformly random points $P$ and $Q$ inside the square.
Draw a line $l_P$ connecting $(0,0)$ and $P$.
Draw another ...
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Probability and average height of two lines intersecting above the x-axis
For context, I know at least basic calculus and what random variables are, but not much about doing calculations with random variables. I came up with a problem for myself where there are two points ...
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A question of random points in a square and probability of intersection of their line segments
The following is a problem from PUMaC 2007:
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square
and draw the line segment from it to $(0,0)$. ...
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Random points in 2-sphere
Say we have the PDF of uniformly picking a random point in the unit $2$-sphere, with distance from origin of the point being $r$, given as
$$
f_R(r) = \begin{cases}
3r^2 & 0 \leq r \leq 1 \\
...
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Probability of line segments intersecting on a plane - A generalization to Buffon's needle problem
I came up with this problem:
If I draw a length 1 line segment randomly, then draw another one,
what's the probability that they'll intersect?
More precisely,
Consider a rectangular area of size $w\...
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Expected number of packs until two cards are collected
Say we want to collect two cards A and B, each appearing independently in the pack with probability $p$. Note that a pack may contain both A and B at the same time with probability $p^2$. I want to ...
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Probability density contour of bivariate normal distribution
I have a bivariate normal distribution and I want to determine the axes of the ellipse that will contain 60% probability. According to my textbok, It follows from the spectral decomposition of the ...
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Conditional probability, two vector valued RVs on a circle.
Given two vector valued RVs $ X,Y \in S_1$ (the unit circle) that are uniformly distributed. We choose a fixed vector $a \in S_1$, e.g. in x-direction: $a=e_x$.
We then sample once from both ...
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Calculate expected number of heads in 10+$\xi$ coin tosses (GRE problem)
This is a problem from a preparatory GRE preparatory GRE test made by guys form University of Chicago.
Problem:
A man flips $10$ coins. With $H$ the number of heads, and $T$ the number of tails, the ...
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Geometric probability: Estimating regions devoid of random points
First, generate $N$ random points in $(0,1)^2$ according to the standard uniform distribution $\mathrm{U}(0,1).$ Then generate and superpose a set of $S$ curves $\phi_S(x)=\exp\bigg(\frac{\log^2 S}{\...
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What is the probability that the distance between $a$ and $b$ is greater than $3$?
Points $(x,y)$ are selected at random where $0\leq x\leq3$ and $−2\leq y \leq 0$. This means that for instance, the chance $(x,y)$ belongs to the square $[1/2,2]\times[−1,0]$ equals $\frac{1.5}{6}$.
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