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.

Geometric probability is a tool to deal with the problem of infinite outcomes by measuring the number of outcomes geometrically, in terms of length, area, or volume. In basic probability, we usually encounter problems that are "discrete" (e.g. the outcome of a dice roll; see probability by outcomes for more). However, some of the most interesting problems involve "continuous" variables (e.g., the arrival time of your bus

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How thick should a cylindrical coin be for it to act as a fair three-sided die?

When flipping a coin of radius $r>0$ and thickness $t>0$ in the real world, there is some non-zero probability of getting neither heads nor tails, but instead landing on the thin lateral side. My question is, how thick does this lateral face need to…
confused_wallet
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A random sphere containing the center of the unit cube

Inspired by a Putnam problem, I came up with the following question: A point in randomly chosen in the unit cube, a sphere is then created using the random point as the center such that the sphere must be contained inside the cube (In other words,…
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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 of the intersection of the two cirlces that…
sys
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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 norm. Is there a similarly simple and clever…
gappy
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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 the following experiment in Mathematica 8: …
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How was "Number of ways of arranging n chords on a circle with k simple intersections" solved?

The problem whose solution is based on the solution to the problem in the title came up as I was trying to find a simpler variant of my needle problem. I we were to uniformly, randomly and independently set $2n$ points on a circle, and then…
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Probability of circle given by randomly chosen diameter falling inside a square

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?
Xxx
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What is average distance from center of square to some point?

How can I calculate average distance from center of a square to points inside the square?
Newcommer
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Expected values of some properties of the convex hull of a random set of points

$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull. For what $N$ do we know the exact values of $P(N)$,…
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Probability Based on a Grid of Lights

The question is as follows : A grid of $n\times n$ ($n\ge 3$) lights is connected to a switch in such a way that each light has a $50\%$ chance of lighting up when switched on. What is the probability that we see a closed curve after turning on the…
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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 a problem of this sort, where the sample space…
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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 suppose I do this again and again, picking points…
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Expected tetrahedron volume from normal distribution

Two equivalent formulas for the volume of a random tetrahedron are given. Further on you can find an interesting conjecture for the expected volume that shall be proved. Tetrahedron volume Given are 12 independent standard normal distributed…
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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 ball: $u \sim U(B(0, 1))$. Then we sample $M$ points…
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Probability - Length of an arc which contains a fixed point

On the circumference $x^2+y^2=1$ one randomly chooses (uniformly and independently) $3$ points. These points split the circumference, forming $3$ arcs. What's the expected value of the length of the arc which contains the point $(1,0)$? Is there any…
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