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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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.
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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?
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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 ...
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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?
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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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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 ...
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Centre in N-sided polygon on circle [closed]
What's the probability that a n-sided polygon made from n distinct random points on circle contain the centre?
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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 ...
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The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.
The vertices of a triangle are three uniformly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
Show that $P(ab>c)=\frac12$.
The result is strongly suggested by ...
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A surprising dilogarithm integral identity arising from a generalised point enclosure problem
This question asked:
What is the probability that three points selected uniformly randomly on the unit circle contain a fixed point at distance $x$ from the circle's centre?
I answered that ...
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Probability that polygon formed by n points on a circle contain the center of the circle?
I have seen similar questions being asked on this forum, but couldn't find this exact problem.
So there are n points selected uniformly randomly on a circle. What is the probability that the polygon ...
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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 ...
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A square contains many random points. From each point, a disc grows until it hits another disc. What proportion of the square is covered by the discs?
A square lamina contains $n$ independent uniformly random points. At a given time, each point becomes the centre of a disc whose radius grows from $0$, at say $1$ cm per second, and stops growing when ...
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Break a stick at two random points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/2$. Why?
Choose two independent uniformly random points on a stick, and break the stick at that those points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/...
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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 ...
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