All Questions
Tagged with geometric-probability calculus
25
questions
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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 ...
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5
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2
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391
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A geometrical puzzle involving calculus
Some time ago I stumbled across a problem from the Putnam Mathematical Competition. I could not find it, but I remember the text quite well.
There are two vectors: a=(10, $y$) and b=($x$,10), where $0 ...
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0
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1
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Integrating $\int_0^1 \int_0^1 |x-y|\,\text{d}x\,\text{d}y$ by hand
Here's a problem from my probability textbook:
If two points be taken at random on a finite straight line their average distance apart will be one third of the line.
What I did: I got the integral$${...
- 1,415
3
votes
3
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Average area of triangle formed ${1\over8}$ that of square
Here's a question from my infamous probability textbook:
A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is ...
- 1,415
9
votes
3
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574
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Probability all angles of triangle formed within semircircle less than $120^\circ$
$3$ points $A$, $B$, $C$ are randomly chosen on the circumference of a circle. If $A$, $B$, $C$ all lie on a semicircle, then what is the probability that all of the angles of triangle $ABC$ are less ...
- 1,415
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2
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510
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Probability that a Particle which moves Unit distance in a Random direction on each step will be inside the Unit Sphere after $n$ steps
The following integral equation arises while calculating the probability that, a particle which starts at the origin and moves a unit distance in a random direction on each ‘move’, will be within the ...
- 14.5k
2
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2
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267
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probability involving two points inside a sphere
Find the probability that two randomly selected points inside a sphere of radius $r,$ are at most $d$ apart, where $0\leq d \leq 2r$.
I've seen several answers on Math Stack Exchange about this, but ...
user747916
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What is the probability that out of $3$ points on an interval $[0,1]$, the first point is closest to an end point on the interval?
So I did this question in $2D$ first and found that out of two points, say $x$ and $y$, the probability that the $x$ is closer to $1$ is half and that the probability that the $x$ is closer $0$ is ...
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1
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Probability that two random disks inside the unit disk have finite intersection
Below, 'circle' refers to a disk.
After seeing the pepperoni pizza problem, I came up with my own question regarding random circles inside circles.
Let $G$ be a circle of unit radius. Suppose we ...
3
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2
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294
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Average angle between two randomly chosen vectors in a unit square
Consider two randomly chosen vectors $(a,b)$ and $(c,d)$ within the unit square, where $a, b, c,$ and $d$ are chosen uniformly from $[0,1]$. What is the expected angle between the vectors?
Here's what ...
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1
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1
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Average distance of two points on a circle
I stumbled upon the question of the average distance of two points on a circle, I learned how to calculate this with polar cordinates (find the distance as a function of $\theta$, itegrate the general ...
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4
votes
1
answer
172
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Randomly choose $n+1$ points on a $S^{n-1}$, probability of $n$-simplex containing center
Randomly choose $n+1$ points on a $S^{n-1}$(surface of ball in $n$-dim space). What's the probability that the $n$-simplex formed by these $n+1$ points contain the center of the sphere?
I conjecture ...
- 1,231
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Answer To A Probability Problem About Placing Random Points In A Circle.
Problem Statement: Three dots are randomly placed in a circle of radius one
cm. What is the probability that when a fourth dot is placed (randomly) in the circle, it is at least one cm away any of the ...
21
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2
answers
3k
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Two individuals are walking around a cylindrical tower. What is the probability that they can see each other?
It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem:
Let there exist some tower which has the shape of a ...
- 602
4
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1
answer
298
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Probability of two fixed-length line segments intersecting within a circular domain
Imagine placing a line segment P of length a on the XY plane such that its middle is at the origin, but its orientation is random (i.e. random angle). Then suppose you placed another line segment Q of ...
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