All Questions
Tagged with geometric-probability circles
21
questions
20
votes
3
answers
574
views
A mysterious limit: probability that a triangle captures the centre of a circle.
On a circle, choose $6n$ $(n\in\mathbb{Z^+})$ uniformly random points and label them $a_0,a_1,a_2,\dots,a_{6n-1}$ going anticlockwise, with $a_0$ chosen randomly.
Draw three chords:
Chord $a_0 a_{3n}$...
- 25.7k
16
votes
1
answer
636
views
A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. The petals are randomly chosen, but they must be ...
- 25.7k
12
votes
1
answer
231
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The vertices of a pentagram are five random points on a circle. Conjecture: The probability that the pentagram contains the circle's centre is $3/8$.
The vertices of a pentagram are five uniformly random points on a circle.
Is the following conjecture true: The probability that the pentagram contains the circle's centre is $\frac38$.
(The ...
- 25.7k
15
votes
2
answers
523
views
The vertices of a hexagon are random points on a unit circle; $a,b,c$ are the lengths of three random sides. Conjecture: $P(ab<c)=\frac35$.
The vertices of a hexagon are uniformly random points on a unit circle; $a,b,c$ are the lengths of three distinct random sides.
A simulation with $10^7$ such random hexagons yielded a proportion of $0....
- 25.7k
9
votes
2
answers
338
views
Probability that Mercury is the nearest planet to Earth.
Motivation: We tend to think of Venus as the nearest planet to Earth because at its nearest approach to Earth, Venus is the closest at 39 million Km away. This is followed by Mars at 56 million Km and ...
- 20.8k
5
votes
1
answer
137
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Conjecture: Two different random triangles (both based on random points on a circle) have the same distribution of side length ratios.
On a circle, choose three uniformly random points $A,B,C$.
Triangle $T_1$ has vertices $A,B,C$. The side lengths of $T_1$ are, in random order, $a,b,c$.
Triangle $T_2$ is formed by drawing tangents to ...
- 25.7k
17
votes
3
answers
953
views
Three random points on $x^2+y^2=1$ are the vertices of a triangle. Is the probability that $(0,0)$ is inside the triangle's incircle exactly $0.13$?
Three uniformly random points on the circle $x^2+y^2=1$ are the vertices of a triangle.
What is the probability that $(0,0)$ is inside the triangle's incircle?
(This a variation of the question &...
- 25.7k
25
votes
2
answers
611
views
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 ...
- 25.7k
3
votes
2
answers
462
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Geometric probability: line segment intersecting a circle?
I'm interested in formulating a $2\text{D}$ geometric probability.
Given:
$(1)$ a circle of radius $r < \frac{1}{2}$ with origin $O$ at the center of a unit square
$(2)$ two points $\{A,B\}$ ...
- 132
0
votes
2
answers
126
views
What is the average radius of the circle containing three random points in a square?
Let $x, y, z$ be three noncollinear points, chosen uniformly and independently from the interior of a unit square. There is a unique circle that passes through $x$, $y$, and $z$. What is the expected ...
- 3,373
1
vote
1
answer
677
views
Average arc length of a uniformly distributed points
Suppose I have $N$ points on a unit circle. I am aware that for a uniform distribution the average angle of one point is not defined, but I want to calculate the average arc length between nearest-...
- 185
0
votes
1
answer
672
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Probabilities With Random Points in a Circle
I'm currently doing some mathematical analysis of a system and I have distilled the problem down to a geometry/probability problem. I have a two part problem I would like to solve.
Part 1: Random ...
- 53
1
vote
0
answers
96
views
Probability of two circles colliding in intersection area of two bigger circles
I have a small circle with area $A_s$ that is bound to be in a bigger circle with area $A_b$. The probability of the small circle being at a specific place in the bigger circle is: $$P = A_s/A_b$$
...
- 11
8
votes
1
answer
958
views
Probability that a random triangle with vertices on a circle contains an arbitrary point inside said circle
What is the probability that a triangle with vertices uniformly randomly distributed on a circle contains the circle's centre?
This is already done (similar to Putnam 1992 A6). The answer is $\frac14$...
- 810
10
votes
1
answer
246
views
What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?
The vertices are chosen completely randomly and all lie on the circumference. Is there a formula for the chance that an $n$-gon covers over $50$% of the area of the circle, with any input $n$?
I ...
- 3,054