3
votes
2answers
42 views

Placing a circle in a square lattice

Two part question. Consider the square lattice $\mathbb{Z}^2$: Imagine you are going to place a circle of radius $r$ somewhere in $\mathbb{R}^2$. Question 1: What is the radius of the largest ...
0
votes
1answer
41 views

fixed length random chord outside of circle.

consider a uniform distribution on a unit circle, I construct a cord by the following steps: pick one endpoint A within the unit circle uniformly. points that are $0<d<1$ distance away from ...
3
votes
2answers
311 views

Expected number of points on circle to form an acute angled triangle

This problem was asked to me in an interview. We keep on adding points on a circle uniformly until there exist three points on the circle which form an acute angled triangle. What is the expected ...
0
votes
1answer
82 views

Finding Area and probability[ hard nut to crack].

Suppose that $X$ and $Y$ are iid uniform distribution with $U(0, 1)$ random variables. (a) What is $\mathbb P((X, Y ) ∈ [a, b]×[c, d])$ for $0 ≤ a ≤ b ≤ 1$ and $0 ≤ c ≤ d ≤ 1$ ? What is $\mathbb ...
1
vote
0answers
68 views

Probability of a certain circular configuration

Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - ...
10
votes
3answers
3k 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?
3
votes
1answer
186 views

Maximum gap among N points on a circle

If $N$ points on the circumference of a circle are chosen at random, what is the probability $F(\theta)$ that the maximum gap between neighboring points is at least $\theta$? Because the gaps sum to ...
5
votes
3answers
2k views

How many right angled triangles can a circle have?

Here's what I recall of the question from CNML Grade 11, 2010/2011 Contest #3, Question 7: There are 2010 points on a circle, evenly spaced. Ford Prefect will* randomly choose three points on ...