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.
638
questions
0
votes
0
answers
18
views
Measure transport by a random matrix
I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are ...
- 1,085
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
77
views
Probability of a random cyclic quadrilateral enclosing a fixed point in its circle
I finally found a single integral solving the natural generalisation of the problem discussed here:
For $n\ge1$ pick $n+2$ points uniformly at random on the unit circle. What is the probability $P_n(...
- 104k
1
vote
0
answers
30
views
Probability of two geometric conditions happening together
I have a problem combining geometry with probability, and I feel like I do not understand the basics to approach this problem.
Let there be two points $\mathbf{p}_1$ and $\mathbf{p}_2$ in $\mathbb{R}^...
- 95
22
votes
4
answers
1k
views
Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and the $x$-axis, from $x=0$ to $x=\pi$.
Where the question came ...
- 25.7k
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
19
votes
4
answers
747
views
Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.
My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation?
Context
Some probability questions have answer $\frac{1}{2}$, and - as ...
- 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
2
votes
1
answer
386
views
The probability of a circle in a circumscriptible polygon
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
why "the area of the polygon $abcelef$
.... represents the number of ways the three ...
- 335
8
votes
2
answers
177
views
Probability of each type of inscribed octahedron
Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. ...
- 5,139
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
16
votes
2
answers
829
views
The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.
The vertices of a triangle are three uniformly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
Show that $P(ab>c)=\frac12$.
The result is strongly suggested by ...
- 25.7k
5
votes
4
answers
263
views
Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square
Question: Suppose that points $P_1$, $P_2$, and $P_3$ are chosen uniformly at random on the sides of a square $T$. Compute the probability that $$\frac{[\triangle P_1 P_2 P_3]}{[T]}>\frac{1}{4}$$ ...
- 690
0
votes
1
answer
45
views
Probability Theory: Generating Functions of Random Variables
Let $X, Y$ be independent random variables with the geometric distribution with parameter
$p > 0$.
(a) Compute the mean of $Z = XY$.
I got that $E(Z) = 1/p^2$
(b) Compute the probability ...
