It's a "dilemma" that I encountered in real life problems. I believe that it's not new and is quite easy to explain, but may look puzzling at first glance.
Alice and Bob are playing with a loop of rope (i.e. a piece of rope in the shape of a circle) of length $1$ meter. The process goes like this:
Firstly, Alice paints the entire loop blue, and then uses a red pen to draw a mark on one uniformly random point on the loop.
After that, Bob takes a pair of scissors and cuts the loop at two independently uniformly random points.
In the end, the loop is cut into two pieces of ropes, with a red mark on one of the two pieces.
Now they start to argue about the expected length of the piece of rope containing the red mark.
Bob:"What I didn't tell Alice is that I'm actually color-blind, hence I could not see the red mark that she drew. Therefore my two random cuts didn't have anything to do with her red mark, and after the two cuts, each piece should have expected length $\frac 1 2$ meter. Whichever piece her red mark belongs to, the expected length is $\frac 1 2$ meter."
Alice:"What I didn't tell Bob is that, before drawing the red mark, I secretly cut the rope at that point and then glued the cut back. I then drew the red mark on exactly the same point. Bob didn't notice that during the whole process. Therefore if I reveal my cut in the end, then the final status should be equivalent to cutting the loop at three independently uniformly random points, and hence the expected distance between any two of the three cutting points is $\frac 1 3$ meter. This means that, when I don't reveal my cut, the expected length of the piece containing the red mark is $\frac 2 3$ meter."
Who is correct, and where did the other make a mistake?