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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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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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 ...
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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 ...
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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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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:...
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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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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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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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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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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 ...
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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 ...
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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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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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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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Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and the $x$-axis, from $x=0$ to $x=\pi$.
Where the question came ...
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