You choose at random ten points inside of a circle, so that no two of them are
on any diameter. What is the probability that the circle has some diameter, so that exactly
five points are on one side and exactly five points are on the other?
Pick:
A) 63/256	B) 1/2	C) 2/3	D) 5/6	E) 1
The randomness is a red herring: There is some diameter dividing the points 5 - 5   no matter what!   Note that none of the points can be in the center of the circle. (Then any diameter going   through any of the points would contain two points.) Choose any diameter. If five points   are on either side, we're done. Otherwise, let n be the larger number of points and rotate   the diameter gradually 180 degrees. As we do this, the number of points on the side with   n changes whenever the diameter crosses a point, eventually dropping to 10 - n. Since 5 is   between n and 10-n, and this number of points changes one at a time, it must be 5 at some point   correct answer 1
<result>E</result>
You choose at random ten points inside of a circle, so that no two of them are
on any diameter. What is the probability that the circle has some diameter, so that exactly
five points are on one side and exactly five points are on the other?