All Questions
Tagged with geometric-probability integration
34
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
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
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
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
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
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
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
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
3
votes
0
answers
175
views
The probability of getting exactly $k$ crossings in buffons needle problem
I am studying Buffons needle problem and I am currently trying to derive the probability of getting exactly $k$ crossings for the situation $l > d$ where $l$ is the needle length and $d$ is the ...
- 1,847
6
votes
1
answer
379
views
Probability of line segments intersecting on a plane - A generalization to Buffon's needle problem
I came up with this problem:
If I draw a length 1 line segment randomly, then draw another one,
what's the probability that they'll intersect?
More precisely,
Consider a rectangular area of size $w\...
- 691
1
vote
0
answers
38
views
The Circle in a Circle problem-Probability that circle C2 passing through 3 randomly selected points in circle C1 lies completely inside C1. [duplicate]
Consider a circle. Three points are chosen at random inside the circle. What is the probability that the circle which passes through these three points while lie totally inside the original circle?
In ...
- 431
1
vote
1
answer
90
views
Probabilistic vs Geometric Theory of Integration
Motivating Question: Let $X$,$Y$ be independent standard uniform random variables. How does one show, rigorously, that
$$
\mathbb{E}[X \mid X+Y = 1] = \frac{1}{2}?
$$
I would be interested in hearing ...
- 1,003
1
vote
1
answer
124
views
An interesting identity about the expectation $\Bbb E(X\mathbb{I}_{\{X<a\}}\mathbb{I}_{\{Y<b\}})$
Given $(X,Y)$ follows a bivariate normal distribution $\mathcal{N}\left(\begin{pmatrix}
0\\
0
\end{pmatrix};\begin{pmatrix}
1 & r\\
r &1
\end{pmatrix} \right)$ with $-1<r<1$
Using ...
- 16.8k
1
vote
0
answers
57
views
Closed-form formula for $u(t):= \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]$, where $(X_1,X_2,\ldots,X_d)$ is uniform on sphere and $h(x):=(x+c)^k$
Fix $c \in \mathbb R$ and an integer $k \ge 0$, and consider the function $h:\mathbb R \to \mathbb R$ defined by $h(x) := (x+c)^k$. Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of ...
- 9,575