$3$ chords are uniformly selected in a circle. We need to find the distribution of number of points of intersection of chords.
To solve this, I first considered an easier problem, one containing only two chords. This was easy to solve and the answer was Bernoulli($1/3$) which I was able to verify using Python.
Next, I assumed each pair of line (in the original $3$ chord problem) must also follow the the above result independently and hence must follow Binomial$(3,1/3)$ i.e. sum of $3$ Bernoulli$(1/3)$ but this was wrong. The actual density was $[5,6,3,1]*1/15$ for $[0,1,2,3]$ reprectively.
Independence was the only assusmption I made so I suspect that must be wrong. However, I dont see why if $C1$ and $C2$ intersect then $C1$ and $C3$ intersecting should have more/less probablity. Any intuition?