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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 thick should a cylindrical coin be for it to act as a fair three-sided die?

When flipping a coin of radius $r>0$ and thickness $t>0$ in the real world, there is some non-zero probability of getting neither heads nor tails, but instead landing on the thin lateral side. My question is, how thick does this lateral face need to…
confused_wallet
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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 probability distribution in the plane, if we know the…
joriki
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Probability that one of a set of four points lies within the triangle formed by the other three

Given four points, each randomly chosen with a uniform probability distribution in the interior of a (WLOG unit) circle, what is the probability that (any) one of the points lies within the triangle formed by the other three. This is (meant to be)…
Doug Couchman
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What's the probability for two points to lie on the same side of the line joining two other points?

While trying to answer this question I realized that the probability for two points to lie on the same side of the line joining two other points is directly related to the probability for four points to form a convex quadrilateral. Since results are…
joriki
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Expected number of disks to fill square

A square with side length $a$ shall be filled with circular disks of radius $r$. In each turn, a new disk is placed randomly inside the square such that it doesn't overlap with any of the previously placed circles. This picture shows an example for…
Simon Bohnen
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Point inside three triangles.

Inside triangle E (the large red triangle) there are 2 smaller similar triangles. A ( the yellow triangle ) and B ( the blue triangle ), both of these smaller triangles have a base length that is 1 / 2 the base of E and a height length that is 1 /…
bobbym
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Probability --- segments

Inside a line segment $E$ with length $6$ unit, there are $2$ segments $A$ with length $2$ unit and $B$ with length $3$ unit. The position of $A$ is fixed with its left end being $k$ unit from the left end of $E$. Thus the …
mrwong
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Geometric Probability - a circle within a circle

In the center of a rounded table with a radius of 50cm, there is a smaller circle with a radius of 10cm. A coin with a radius of 1cm is thrown on the table. Assuming that it will always land on the table, what is the probability that the entire coin…
Jankel
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Probability that $a+b+c+d = 1.5$ if $a+2b+3c+4d = 3.5$

Let $(x_1, x_2, x_3, x_4) \in [0,1]^4$ be a vector chosen uniforly at random. What is the probability that we observe? \begin{eqnarray*} x_1 + 2 x_2 + 3 x_3 + 4 x_4 &=& \;\;\,3.5 \pm 0.01\\ x_1 + \;\,x_2 +\;\, x_3 +\;\, x_4 &=& 1.5 \pm 0.01…
cactus314
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Expectation of overlapping triangles

Let E denotes a triangle PQR with PQ = QR = 2 unit and angle Q = 90 degree . L , M and N are mid-points of PQ , QR and RP respectively . Let A denotes triangle PLN , B denotes triangle LQM , C denotes triangle NMR and X denotes triangle LMN . If A…
mrwong
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Random points in a cube.

A point with coordinates $x$,$y$,$z$, is chosen uniformly at random from a cube: $$\{(x,y,z)\in \mathbb{R^3}:0\le x,y,z \le 10\}.$$ Assume that the probability of an event is proportional to the volume of a cube. What is the probability of the…
user91689
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Expected area of triangle inscribed in a circle

On a unit circle , BC is a chord with length 4/π . Point A is picked randomly at its circumference . Find the expected area of △ ABC .
mrwong
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Difference of numbers in a unit interval

$x,y,z\in \mathbb{R}$ are chosen at random from the unit interval $[0, 1]$. What is the probability that $$\max(x,y,z) - \min(x,y,z) \leq \frac{2}{3}$$ EDIT- Solutions not using calculus would be appreciated, as this problem appeared on a test…
K. Chopra
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Geometrical probability - distance in the interval

We randomly (uniformly) generate points $[a,b]$, where $0 \leq a,b \leq 1$. Let $\epsilon$ be a given real number from the interval $[0, \sqrt{2}]$. What is the probability that if we generate two points $P_1,P_2$, then their distance will be at…
Patrik Bak
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We take two points X and Y on the interval (0, 2). What is a probability of the event A = {X + $Y^2$ > 0.1}?

I try to solve problems from my textbook, but I am stuck on this problem. We take two points X and Y on the interval (0, 2). What is a probability of the event A = {X + $Y^2$ > 0.1}? My approach to solve this problem is to modify inequality, so I…
Janey
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