Medium Difficulty Math Problem

Two points (A and B) will be chosen at random on a circle. A chord will be drawn between the two points. What is the probability that the chord's length is less than that of the radius? (Basically, what is the probability that AB<r). I came up with this on my own and solved it on my own. Although I'm sure someone has done both of those things before.

1/3 I believe

yes

This reminds me of Bertrand paradox a bit! https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

BONUS QUESTION:

Find a function that returns the probability that the distance between two random points on a circle are larger than x

Where x is a fraction of the radius between 0 and 1

Examples:

f(0) =

f(1/2) =

f(1) = 0

I'm bad at formalizing questions, can someone who understands what I mean help me out?

Took me awhile, I got f(x) = arccos(1-x)/pi

I actually arrived to f(x)= 1- 2/pi arcsin(x) (1 ≥ x ≥ 0) with $x$ interpreted as a fraction of the diameter, not the radius.

I don't quite get the problem

Suppose two points are chosen independently and uniformly at random on the circumference of a circle of radius $R$. Let $d$ be the straight-line distance (the chord length) between the two points. For a number $x$ between 0 and 1, define f(x) to be the probability that $d$ is greater than $xR$.

In other words:

f(x)=Pr⁡(d>xR),0≤x≤1.

For example:

• f(0)=1

• f(1/2)=?

• f(1)=0

Youre supposed to find an explicit formula for f(x).