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 the tag [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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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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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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A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?

A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random points and $7$ clusters, with an average cluster size…
Dan
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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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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 probability that $ac > b^2$ is $\displaystyle…
Nilotpal Sinha
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My student's wrong method gives the right answer? A question about random points on a circle.

I came up with the following question. Four uniformly random points on a circle are chosen, and line segments are drawn between each pair of points. What is the probability that the longest line segment is between neighboring points? Exerimental…
Dan
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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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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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Prove $\int_0^{\sqrt2/4}\frac1{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)(x-1+x\sqrt{9-16x})}{1-2x}}dx=\frac{\pi^2}{8}$ (from a probability question)

Let $$I=\int_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)(x-1+x\sqrt{9-16x})}{1-2x}}dx$$ Prove that $I=\dfrac{\pi^2}{8}$. Wolfram suggests that it's true but does not find the antiderivative. Here is the graph of the function being…
Dan
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Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.

My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation? Context Some probability questions have answer $\frac{1}{2}$, and - as you might expect - have an intuitive…
Dan
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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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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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