I find this easier to visualize with an equivalent way to pick points: drop four (n) lines through the center of the circle. For each line, randomly pick one of the two points that intersect the circle.
Independent of the lines dropped, there are eight (2n) ways to pick adjacent points and sixteen (n^2) different combinations of points.
There are four (n) adjacent points iif the points lie on the same half of the circle (proof by interactive visualization).
Independent of the lines dropped, there are eight (2n) ways to pick adjacent points and sixteen (n^2) different combinations of points.
There are four (n) adjacent points iif the points lie on the same half of the circle (proof by interactive visualization).
So the answer is eight sixteenths (2n/2^n).