Q:

Probability that two triangles inscribed in a circle do not intersect.

Suppose three random points are picked on a circle, and the points connected to form a triangle. This is repeated. What is the probability that the two triangles don't intersect?
I think there's a way to do this by thinking of the triangles as two arcs formed by the three points, that don't intersect eachother, but that also seems difficult.

A:

You made a great observation that "these triangles intersect iff their arcs intersect"
Hint: Use the Law of Iterated Expectation.
For clarity, let's form a red triangle and a blue triangle.
Given 6 points on the circle, there are ${ 6 \choose 3  }$ ways to form a red triangle and blue triangle.
(Changing the colors count as a distinct ways. Choosing 3 red points is sufficient to determine the triangles)
Out of those, exactly 6 of those ways do not intersect.
(The 3 red points have to be consecutive.)
Hence, the probability is

 $\frac{ 6}{20} $

