All Questions
Tagged with geometric-probability combinatorics
21
questions
1
vote
0
answers
78
views
Average number of hyperquadrants in a random subspace
Suppose I have a random $n$-dimensional linear subspace of $\mathbb{R}^m$. How many of the $2^m$ hyperquadrants does this space intersect, on average? Alternatively, what are the odds that this ...
- 1,629
1
vote
0
answers
70
views
Prove there are at least $2$ lines with the property that each of them divides the plane into $2$ regions with the same number of red and blue points. [duplicate]
Let $n \geq 2$ a natural number, and $2n$ points chosen in plane, $n$ red points and $n$ blue points. There are no $3$ collinear points among the $2n$ points in plane. A 'good' line is a line that ...
1
vote
1
answer
58
views
Paratrooper falling in a 4km^2 squared field with trees, probability of him not getting stuck
paratrooper is falling in a square field with a side of 2km.
in each corner of the field there is a big tree.
the paratrooper gets stuck in the tree if he falls within a 1/11 km distance of the tree.
...
- 19
0
votes
0
answers
60
views
probability of five people meeting at train station
Five people all go to the same train station on their way to work. Each of them will arrive at a random time from 7:00 AM to 9:00 AM and stay for exactly 12 minutes, independent of all the others. ...
- 21
0
votes
1
answer
110
views
Probabilistic geometric problem in high dimensions
In the $n$-dimensional Euclidean space, let $H$ be a random hyperplane selected in way similar to the so called "random radial point" (used to propose a solution for the Bertrand paradox): ...
- 149
2
votes
1
answer
259
views
Geometric probabilistic problem on a plane
Let $T$ be a triangle with vertices belonging to a given Cartesian plane $P$ and with side lengths $a$, $b$ and $c$, where $a\ge b\ge c\ge 0$. Let $L_T$ be a straight line selected uniformly at random ...
1
vote
1
answer
95
views
Given $372$ points in a circle with a radius of $10$, there is an annulus with radii $2$ and $3$ containing at least $12$ of these points.
Given $372$ points in a circle with a radius of $10$, show that there is an annulus with inner radius $2$ and outer radius $3$, which contains not less than $12$ of the given points.
My thinking is ...
- 43
2
votes
1
answer
107
views
Number of random unit vectors which are less than theta apart
Given $n$ unit vectors which are uniformly distributed on a unit sphere, what the expected number of groups of $k$ vectors which are within an angle $\theta$ of one another
For example, if I have $n=...
- 355
0
votes
0
answers
62
views
Expected number of hyperplane cuts to partition a $d$-dimensional hypercube of side length $n$ into pieces of unit volume
I came up with the following question and am not certain of its solution. Has this problem or problems like it be tackled before?
Let $H_d$ be the $d$-dimensional hypercube $[0,n]^d \subset \mathbb{...
- 303
12
votes
1
answer
583
views
Probability Based on a Grid of Lights
The question is as follows :
A grid of $n\times n$ ($n\ge 3$) lights is connected to a switch in such a way that each light has a $50\%$ chance of lighting up when switched on. What is the probability ...
- 14.5k
1
vote
0
answers
170
views
generalization of Dyck Path: size K upward steps
One of the many interpretations of Dyck Paths is the number of lattice paths from $(0,0)$ to $(n,n)$, staying at or below the diagonal $y=x$, using only 2 kinds of line segments (1 unit right, or 1 ...
- 645
6
votes
3
answers
3k
views
Two diagonals of a regular nonagon (a $9$-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?
Two diagonals of a regular nonagon (a $9$-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?
This is similar to a previous problem that I posted ...
- 953
1
vote
2
answers
67
views
Easier Way to Find Probability
I know how to compute this with a concept similar to truth tables. First I listed all of the combinations of the angle in trios:
$ABC$, $ABD$, $ABE$, $ACD$, $ACE$, $ADE$, $BCD$, $BCE$, $BDE$, $CDE$.
...
- 269
15
votes
1
answer
607
views
How was "Number of ways of arranging n chords on a circle with k simple intersections" solved?
The problem whose solution is based on the solution to the problem in the title came up as I was trying to find a simpler variant of my needle problem.
I we were to uniformly, randomly and ...
- 12.5k
1
vote
2
answers
177
views
What is the average number of balls one should extract until all three colours have appeared?
Suppose an urn contains a large number of balls. 50% of the balls are black, 30% are white and 20% are red. Balls are extracted randomly, one at a time, recorded and returned to the urn. What is the ...
- 197