Random chord geometry

We have 8 points on a circle, so there are $\binom{8}{2}=28$ chords in total. Choosing 4 chords uniformly at random gives $\binom{28}{4}$ equally likely outcomes.

For the 4 chords to form a convex quadrilateral, their union must be exactly the boundary of that quadrilateral. Because the points lie on a circle, any 4 distinct points determine a convex quadrilateral whose sides are the chords connecting consecutive points in cyclic order. Thus a favorable set of 4 chords corresponds precisely to choosing 4 of the 8 points and taking the four boundary chords of that set. There are $\binom{8}{4}=70$ ways to choose the points, and each yields exactly one favorable 4‑chord set. No other set of 4 chords can form a convex quadrilateral, since any quadrilateral must have four distinct vertices, and on a circle the only non‑self‑intersecting 4‑cycle on four points is the boundary cycle.

Hence the probability is