# Tagged Questions

71 views

### Expected area of a random triangle with fixed perimeter

I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle ...
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### Find probability of angle being obtuse

We are given points A and B on the 2D plane and distance between them is 2. Let C - randomly picked point on the circle with radius R and center at the middle of AB. Find probability of angle ABC ...
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### Bertrand paradox Random midpoint

http://en.wikipedia.org/wiki/Bertrand_paradox_(probability) The link above explains Bertrand paradox in probability. In "Random Midpoint method" Bertrand uses a concept that all chords whose ...
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### What is the expected size of the convex hull of $n$-points selected randomly in a $2d$-circle?

We know $n>2$ and worst case its a triangle, best case the points all lie on a circle. Can we generalize to higher dimensions? What's the probability that the size of the convex hull of $n$ points ...
574 views

### Expected area of the intersection of two circles

If we pick randomly two points inside a circle of radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the ...
159 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 ...
315 views

### What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
280 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 ...
363 views

### Probability of circle given by randomly chosen diameter falling inside a square

Two dots are thrown into a square with side length 1 cm. The line ending in these two dots is the diameter of a circle. What is the probability that the circle lies in the square?
914 views

### Distance between $2$ random points in a segment.

On a straight line of length $10$ cm, two points A, B are selected at random uniformly and independently. What is the probability that the distance $AB > 4$ cm? Edit: I edited the question to make ...
308 views

### Average area of choosing three points on a surface?

Assume I choose three random points on the surface of a sphere. What is the average area? (Each point is independently chosen relative to a uniform distribution on the sphere) Also, what would be the ...
237 views

### Probability on an infinite plane

(I thought this is a popular problem, but sadly Google yields nothing.) Three points are chosen at random on an infinite plane. What is the probability that they are on a line? And a variant: the ...
235 views

### What is the probability that a quadrilateral is convex?

Given $4$ distinct randomly chosen points $x_1$, $x_2$, $x_3$, and $x_4$ in the plane such that the polygonal path from $x_1$ to $x_2$ to $x_3$ to $x_4$ to $x_1$ describes a non-self-intersecting ...
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### 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 ...