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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How can I find the PDF of this function of normal variables? Or what is the distribution of distances between two random points on a unit sphere?

How can I find the probability density function of the random variable $D = \frac{\sqrt{\left(x-\sqrt{x^2+y^2+z^2}\right)^2+y^2+z^2}}{\sqrt{x^2+y^2+z^2}}$ If x, y, and z are all independently standard normally distributed? I want to know this…
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What is the probability of uniformly sampling a point in d-dimensional hypercube?

Let us consider a hyper-cube whose length is l units along each of its d-dimensional structure. It is desired to uniformly sample a point inside the hyper-cube. How to do uniform sampling and what could be its probability?
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Expected area of an internal triangle determined by a random point in a triangle

A point M is chosen at random (uniformly) inside an equilateral triangle ABC of area 1. How to find the expected area of the triangle ABM?
Koncopd
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Probability of segment lying in circle

Given a circle of radius R: $x^2+y^2\le R$, find probability of horizontal segment with length $\frac{R}{2}$ lie whole inside this circle. Position of segment's center has uniform distribution in circle. Okay, I draw circle for $R=1$: Red circle is…
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Geometric probabilities with rectangle

One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than 1.15? I think that it could be:…
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Probability of two points being a certain distance apart on a circle

Is the probability of two points being a certain distance $k$ apart on a circle of length $m$, with $0\le k<\frac{1}{2}m $, always the same for any $k$?
tfaod
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Three points $x,y,z$ are chosen at random on the point interval $(0,1)$. What is the probability that $x \le y \le z$?

I have been trying this problem for a while. But, couldn't find any solution. How do I solve this?
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Geometric probability -- line segment

Fellow math lovers! It's been quite sometime since I have solved basic probability problems. I am now trying to remember how to calculate geometric probability. As far as I remember, the general formula for a geometric probability calculation is (in…
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probability of quadratic polynomial

How can we calculate the probability of quadratic polynomial such that \begin{align} \ Y=a_0+a_1x+a_2x^2\\ \mathsf P\left(Y\lt y\right)=P \end{align} with y is such a number from this polynomial and is given thanks in advance
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5 independent traffic lights, how many is car expected to pass without getting stopped

$\newcommand{\E}{\mathbb{E}}$ I can't wrap my mind around this one. I keep thinking it is geometric probability problem, but can't get correct solution (which is $\E(X) = 0.6598)$. Problem : There are 5 independent traffic lights, each with chance…
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Geometry Probability Question

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Hi everyone I found this interesting question; help is appreciated! :) We put 15 points on a circle O equally spaced.…
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