Duck Pond

Let the first person be placed at A. O is the centre of the circle. The second person B placed anywhere between BAB' will be less than distance R from A. < BOB' is 120 degrees. Therefore probability that person B is greater than distance R from A is (360-120)/360=240/360=2/3.

We know that the length of the circumference is 2(pi)R, so we have to calculate the probability that if the person is standing anywhere, then the second person has to be R distance apart from the first person's right and left side, along the circumference. This covers the length of the circumference that is 2(pi)R minus 2R, which means that our probability will be represented by [2(pi)R - 2R]/2(pi)R.

Since the solution options are different enough, we can guess pi to be 3.14 and from there calculate that this fraction is 0.6815, which is close to 2/3.

It should have been stated explicitly that we don't consider the distance along the circumference! Yet if we have a look at the offered answers, we came to a conclusion, $R$ is measured along a straight line. When we randomly choose one person, a distance of the length $R$ of the other person from the first one is taken in two opposite directions, which makes $2R$ . Since $R$ is the radius of the circle, the whole length of distances is $6R$ (because we deal with a hexagonal), i.e. the probability is $\dfrac13$ . However, there are two persons, so that the total probability is $2\cdot\dfrac13=\dfrac23$ .