All Questions
Tagged with geometric-probability area
9
questions
2
votes
1
answer
357
views
Probability of picking 2 numbers between 0 and 1 to be within 1/2 distance of each other?
Problem: What's the probability of picking 2 numbers, x & y, between 0 and 1 such that they will be within the distance of $\frac{1}{2}$ of each other?
In other words, $\Pr(\text{distance between ...
- 608
5
votes
1
answer
310
views
What is the expected area of a cyclic quadrilateral inscribed in a unit circle?
Choose four points randomly on the circumference of a circle with radius $1$. Connect them to form a quadrilateral. What is the expected area of this quadrilateral?
I have attempted to simulate to ...
- 1,024
0
votes
1
answer
109
views
What is the probability that a random point would lie in this inscribed circle?
Take the unit square and inscribe a circle $Q$. Then take one of the corners (say the upper-left one) and inscribe a circle $Q'$ in that space (so the circle touch the square twice and $Q$ once). Let $...
- 37.2k
8
votes
1
answer
122
views
Expected area of a random $n$-gon
Choose $n$ points $\{z_1, \ldots, z_n\}$ from the unit circle $\partial \mathbb{D} = \{z \in \mathbb{C}: |z| = 1\}$ uniformly at random, and let $P_n$ be the convex hull of the $z_i$'s. Let $X_n = ...
- 701
7
votes
1
answer
347
views
Area bounded by $\cos x+\cos y=1$
What is the area of the region $\cos x+\cos y > 1$, where $|x|,|y|<\pi$?
In other words, is there a "closed" form -- using functions that are well-known and nice to work with -- for this ...
- 27k
10
votes
1
answer
246
views
What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?
The vertices are chosen completely randomly and all lie on the circumference. Is there a formula for the chance that an $n$-gon covers over $50$% of the area of the circle, with any input $n$?
I ...
- 3,054
3
votes
1
answer
288
views
Numbers $\alpha$ and $\beta$ are selected from interval $[0,1]$. What is the probability that $x^2+\alpha x + \beta ^2=0$ has real roots?
I know that discriminant must be greater than zero , so we have :
$\alpha ^2-4\beta^2\geq 0$
$\alpha^2\geq4\beta^2$
$\alpha\geq 2\beta$
We draw a function $\alpha - 2\beta = 0 $ and our condition ...
user258209
2
votes
3
answers
343
views
Infinite points on a paper?
I remember solving questions like this: On a paper with dimensions $30cm$ x $21cm$ if a rubber (erasers)* is dropped, what is the probability that it falls over a grey shaded region of dimensions $1cm$...
- 1,664
0
votes
1
answer
260
views
Expected area of an internal triangle determined by a random point in a triangle
A point M is chosen at random (uniformly) inside an equilateral triangle ABC of area 1. How to find the expected area of the triangle ABM?
- 995