circlesintersection-theoryprobability theorystatistics
I understand that for 2 random chords, the probability of no intersections is 1/3 thanks to this blog post.
What happens when I have 3 random chords? Is there an intuitive explanation for calculating the number of intersections given n random chords?
I found a post that proposes the following formula for r chords, but this does not hold for 2 chords. Any ideas?
$$P(NoIntersection) = \frac{2^r}{(r+1)!}$$
Note: By random chord, I mean by randomly picking 2 points that lie on the circle.
HINT:-
Take a circle, let its radius be $r$.
Take any point in the circle. It doesn't matter what point you take for the first point.
Now as soon as you choose the second point, you have a chord. Let the two points of the chord make angle $2\theta$ at centre. So the length of the chord is $2\cdot r\cdot \sin\theta$.
Now the given constraint is $\frac{2}{3}\le2\sin\theta \le \frac{5}{6}$.
Now this can be solved easily.
The “good” region for $\omega$ in your solution doesn’t look quite right to me. An equation of the line is $x\cos{\pi\beta}+y\sin{\pi\beta}=2\alpha-1$, with $x$- and $y$-intercepts $(2\alpha-1)/\cos{\pi\beta}$ and $(2\alpha-1)/\sin{\pi\beta}$, respectively. For these the have the correct signs, the point must be in the second quadrant, so we must have $1/2\lt\alpha\le1$ and $1/2\lt\beta\lt1$. The region is symmetric, but not about the line $\alpha=\beta$, as we shall soon see.
Not all values of $\alpha$ in this range produce intercepts in $(0,1]$, which is what we need for the chord to intersect both coordinate axes. For a fixed $\beta$, the maximal value of $\alpha$ is the least value for which the line passes through either the point $(-1,0)$ or $(0,1)$, so $$\alpha\le\min\left(\frac12(1-\cos{\pi\beta}),\frac12(1+\sin{\pi\beta})\right),$$ or equivalently, $$\begin{cases}\alpha\le\frac12(1-\cos{\pi\beta}), &\text{if }\frac12\lt\beta\le\frac34 \\ \alpha\le\frac12(1+\sin{\pi\beta}), &\text{if }\frac34\lt\beta\lt1.\end{cases}$$ The region looks approximately like this:
(the horizontal axis represents $\alpha$).
This region is symmetric about the line $\beta=3/4$, so its area is equal to $$2\int_{3/4}^1\sin{\pi\beta}-\frac12\,d\beta.$$
Best Answer
The first linked blog post shows that the probability that two random chords do intersect is $\tfrac 13$, which means the probability they don't intersect is $\tfrac 23$. This lines up with the formula provided in the second link.