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.

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36
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4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
12
votes
4answers
6k views

What is the probability that the center of the circle is contained within the triangle?

Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?
14
votes
3answers
2k views

Simulating uniformly on $S^1=\{x \in \mathbb{R}^n \mid \|x\|_1=1\}$

A scheme to generate random variates distributed uniformly in $S^2=\{x\in \mathbb{R}^n \mid \|x\|_2=1\}$ is well known: generate a standard normal variate in $\mathbb{R}^n$ and normalize it to unit ...
14
votes
2answers
2k views

Probability that the convex hull of random points contains sphere's center

What is the probability that the convex hull of n+2 random points on n-dimensional sphere contains sphere's center?
17
votes
1answer
355 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
8
votes
1answer
244 views

How is the number of points in the convex hull of five random points distributed?

This is about another result that follows from the results on Sylvester's four-point problem and its generalizations; it's perhaps slightly less obvious than the other one I posted. Given a ...
44
votes
3answers
1k views

Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...
23
votes
5answers
11k views

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
4
votes
2answers
856 views

probabilty of random points on perimeter containing center

related question: probablity of random pick up three points inside a regular triangle which form a triangle and contain the center What is the probability that a (possibly degenerate) triangle made ...
2
votes
1answer
200 views

Expected area of the intersection of of triangles made up random points inside a circle, all the triangles must contain the origin

How to find the expected area of the intersection of a set of triangles made up $N$ random points that are picked uniformly inside a circle? The triangles must contain the origin of the circle. If ...
6
votes
0answers
216 views

Expected Number of Convex Layers and the expected size of a layer for different distributions

It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
3
votes
4answers
736 views

Random walking and the expected value

I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me. Lets say we have a square, with vertex's 1234. I can randomly walk to each ...
9
votes
5answers
621 views

Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary

Background : I happen to love solving tough problems. Problem is, I simply cannot answer some! It happened again today, as I attempted to solve the questions in this site: http://www.skytopia.com/...
19
votes
4answers
3k views

probablity of random pick up three points inside a regular triangle which form a triangle and contain the center

what is the probablity of random pick up three points inside a regular triangle which form a triangle and contain the center of the regualr triangle the three points are randomly picked within the ...
7
votes
1answer
120 views

Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the ...
2
votes
1answer
199 views

Three points on sides of equilateral triangle

Let's choose three points on the sides of an equilateral triangle(one point on each side) and construct a triangle with these three points. what is the probability that area of this triangle be at ...
6
votes
0answers
109 views

Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
4
votes
1answer
1k views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
2
votes
3answers
68 views

Do the lengths of all three segments necessarily have the same distribution?

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have ...
10
votes
2answers
2k views

Expected area of the intersection of two circles

If we pick randomly two points inside a circle centred at $O$ with radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area ...
4
votes
1answer
281 views

What's the probability for two points to lie on the same side of the line joining two other points?

While trying to answer this question I realized that the probability for two points to lie on the same side of the line joining two other points is directly related to the probability for four points ...
3
votes
1answer
55 views

Number of vertices of a random convex polygon

Take $n>2$ random points, chosen independently with uniform probability on $[0,1]\times[0,1]$. What is the probability $P(n,k)$ that the convex hull of these points is a polygon with exactly $2<...