Mathematics Stack Exchange
Mathematics 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.

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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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw the picture and I drew a smaller (concentric)…
terrace
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The "pepperoni pizza problem"

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely randomly. All pepperoni have the same radius $r$. A…
Ben S.
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Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is nearest to him. (A person can get shot more than…
John Smith
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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.
NECing
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Probability that 3 points in a plane form a triangle

This question was asked in a test and I got it right. The answer key gives $\frac12$. Problem: If 3 distinct points are chosen on a plane, find the probability that they form a triangle. Attempt 1: The 3rd point will either be collinear or…
Serenity
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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 fall, can only land in such a way such that they…
Nicolás Kim
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What is the probability that the center of the circle is contained within a triangle formed by choosing three random points on the circumference?

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?
Paul
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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 regular triangle and then form a new triangle and…
zinking
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Matching red to blue dots

I have two red points, $r_1$ and $r_2$, and two blue points, $b_1$ and $b_2$. They are all placed randomly and uniformly in $[0,1]^2$. Each dot points to the closest dot from another colour; closest is defined wrt the Euclidean distance. We use $x…
fox
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Two individuals are walking around a cylindrical tower. What is the probability that they can see each other?

It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem: Let there exist some tower which has the shape of a cylinder and whose radius is A. Further, let…
Fine
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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?
Grigory M
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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 convex pentagon is $\frac{49}{144}$, what is the…
kejma
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Probability of random sphere lying inside the unit ball

Let $n\geq2$. Let $B\subseteq\mathbb{R}^n$ be the unit ball. Randomly choose $n+1$ points of $B$ (uniformly and independently). Then (almost surely) there will be a unique hypersphere $S$ passing through all $n+1$ points. What is the probability…
Thomas Browning
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PDF of volume of tetrahedron with random coordinates

Question What is the probability distribution function (PDF) of the absolute volume of a tetrahedron with random coordinates? The 4 random tetrahedron vertices in $\mathbb{R}^3$ are $$ \mathbf{\mathrm{X}_1} =(x_1^1,x_1^2,x_1^3),\;\; …
granular bastard
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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 cardinality) of a subset in which the points form the…
kejma
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