# 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.

343 questions
1answer
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### Random Variable Y following uniform distribution with parameter Random X that follows geometric.

Random variable X follows geometrical distribution with p=1/4. Random variable Y follows uniform distribution in [-X,X]. I'm looking for P(Y>3/2) and also P(X=2|Y>3/2).I know for a fact that Σ(from k=...
1answer
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### Geometric Probability. Sphere

Suppose a sphere with radius $R$. Find the probability of the event such that $n$ selected points of the sphere are within the distance of $r = \frac{R}{2}$ from the center of the sphere. The points ...
1answer
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### Average area of the shadow of a convex shape [closed]

What is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be ...
1answer
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### How to sample uniformly from the surface of a (fish-) bowl?

Define a fish-bowl as a sphere comprised between two horizontal disks. That is, a sphere where we have replaced the top and bottom sectors by horizontal disks. See picture below. How to sample ...
2answers
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### What is the probability that the angle is a right angle?

Let's say you have a square, say $ABCD$, with side length 1. A point $P$ is randomly chosen from inside the square, with any point being equally probable of being chosen. What is the probability that ...
1answer
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### Expected value of number of dots in random placed circle

Given a square with vertices at $(0,0), (1,0), (0,1), (1,1)$ and $N$ labeled dots in this square with coordinates $(x_i, y_i)$. Bowser picks a dot $(a, b)$ in square (with uniform distribution) and ...
2answers
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### Special case of Bertrand Paradox or just a mistake?

I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
1answer
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### Randomly choose $n+1$ points on a $S^{n-1}$, probability of $n$-simplex containing center

Randomly choose $n+1$ points on a $S^{n-1}$(surface of ball in $n$-dim space). What's the probability that the $n$-simplex formed by these $n+1$ points contain the center of the sphere? I conjecture ...
0answers
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### Method to build a polyhedral die with given probabilities

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
2answers
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### What is wrong with my method for probability that n points on a circle are in one semicircle

So I understand the method used in this solution, and I know my method is incorrect, but I was just looking for an explanation why. I was thinking that if I choose any spot on the circumference, ...
1answer
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### Which cell in a grid a point belongs to?

I have a square which is divided into an NxN grid and I have a dot. How I can find which cell this point belongs to? Assuming I know the boundaries of the square. How to find the point belongs to the ...
0answers
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### What's the probability that equation $x^2 + px + 6.9q = 0$ has real solutions

In the square $K = \{(u,v): u,v \in [0,\;9.6]\}$ randomly selected point with coordinates $(p,q)$. What is probability that equation $x^2 + px + 6.9q = 0$ has real solutions? I was trying to solve ...
1answer
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### What's the probability that triangle area will be smaller than 7.25

Rectangle edges equals a=2.9 and b=6.3. In adjacent rectangle edges randomly selected two points and straight line drawn through them. What is the probability that drawn triangle area is smaller ...
2answers
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0answers
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