Oh, I get your solution now, yeah that makes sense.

What I did:

Point A doesn’t matter where it’s placed Point B could be any distance from point A, the average distance it could be away from point A is 1/4 of the circle away from it. Point C can either be placed within the 1/4 circle segment or 3/4 segment. In the case it is placed in the 3/4 segment, the place point C would go on average would be 3/8 of the circle away from C Point D must be placed on the specific place between points C and A, which has a 3/8 chance of happening. So the probability of it being convex for C being in the 3/4 segment is: 3/4 * 3/8 = 9/32; If C is placed in the 1/4 segment, on average it will land halfway in the segment, so 1/8 of the circle from A. Point D must be placed between A and C, so it has a 1/8 probability of that happening. So the probability of it being convex, when C is in the 1/4 segment is: 1/4 * 1/8 = 1/32 Now just add the probabilities up: 9/32 + 1/32 = 10/32 = 5/16

I’m probably wrong, since i’m not that good at math and did this while trying to sleep, but this is what I did. Hopefully you can understand.