All Questions
Tagged with geometric-probability geometry
165
questions
9
votes
1
answer
271
views
A probability involving side lengths of a random triangle on a disk: Is it really $\frac37$?
Choose three uniformly random points on a disk, and let them be the vertices of a triangle. Call the side lengths, in random order, $a,b,c$.
What is $P(a^2<bc)$ ?
A simulation with $10^7$ such ...
- 25.7k
12
votes
1
answer
231
views
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
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
views
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
10
votes
4
answers
305
views
Draw tangents at 3 random points on a circle to form a triangle. Show that the probability that a random side is shorter than the diameter is $1/2$.
Choose three uniformly random points on a circle, and draw tangents to the circle at those points to form a triangle. (The triangle may or may not contain the circle.) For example:
What is the ...
- 25.7k
20
votes
3
answers
590
views
Probability that the centroid of a triangle is inside its incircle
Question: The vertices of triangles are uniformly distributed on the circumference of a circle. What is the probability that the centroid is inside the incricle.
Simulations with $10^{10}$ trails ...
- 20.8k
6
votes
2
answers
157
views
Probability that the coefficients of a quadratic equation with real roots form a triangle
Question: What is the probability that the coefficients of a quadratic equation form the sides of triangle given that it has real roots? Assume that the coefficients are uniformly distributed and ...
- 20.8k
1
vote
1
answer
50
views
Find the best ratio between grid size and the square size
Let's say I have a bunch of squares of side x and the grid of square sectors, each of side y.
I am placing the squares randomly in this grid - the sides of squares are parallel to sides of the grid, ...
- 61
1
vote
0
answers
112
views
Expected area of inscribed triangle
Three points are uniformly thrown on a circumference of circle of radius 1, find the mathematical expectation of the area of the triangle formed by them.
I've tried to use that formula: $S = \frac{1}{...
- 11
0
votes
0
answers
50
views
$n\times n$ grid filled with $n$ colors. What is the average group size as $n\to\infty$
Take a grid with dimensions $n\times n$ squares and randomly fill each square with $1$ of $n$ colors. What is the expected average group size of colors touching each other as $n$ approaches $\infty$? ...
- 1,688
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
43
votes
3
answers
4k
views
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 ...
- 25.7k
14
votes
4
answers
535
views
Break a stick at two random points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/2$. Why?
Choose two independent uniformly random points on a stick, and break the stick at that those points. The probability that the longest piece is at least twice as long as each of the other pieces is $1/...
- 25.7k
5
votes
2
answers
516
views
What is the probability that the triangle formed by three uniformly random points on the sphere contains its circumcentre?
In answering Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$. it turned out that if you choose three ...
- 239k