TITLE: Probability question about chords on a circle
QUESTION [5 upvotes]: Four points are chosen independently and at random on a circle. Find the probability that chords X1X2 and X3X4 intersect a) without calculation using a symmetry argument and b) from the definition by an integral
I'm lost here. I'm thinking of using some kind of angle argument, but not sure how to go about it.

REPLY [5 votes]: If $\alpha\in[0,2\pi]$ is the angle from $X_1$ to $X_2$, then the probability that $X_3$ is between $X_1$ and $X_2$, and $X_4$ is between $X_2$ and $X_1$, which is one of the configurations were the chords intersect, will be $\dfrac{\alpha}{2\pi} \cdot \dfrac{2\pi-\alpha}{2\pi}$.
Averaging this for all (equiprobable) values of $\alpha$, we find
$$\dfrac{\int_0^{2\pi} \dfrac{\alpha}{2\pi} \cdot \dfrac{2\pi-\alpha}{2\pi} d\alpha}{2\pi} = \left.\left(\dfrac{2\pi\dfrac{\alpha^2}2-\dfrac{\alpha^3}3}{(2\pi)^3}\right)\right|_0^{2\pi}=\dfrac16$$
The probability that $X_4$ is between $X_1$ and $X_2$, and $X_3$ is between $X_2$ and $X_1$ will also be $\dfrac16$, so the probability that there is an interesection will be $2\times\dfrac16=\dfrac13$, consistent with the intuitive answer given by a symmetry argument.