1996 AHSME Problems/Problem 22
Four distinct points, , , , and , are to be selected from points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord intersects the chord ?
Let be a convex cyclic quadrilateral inscribed in a circle. There are ways to divide the points into two groups of two.
If you pick and , you have two sides of the quadrilateral, which do not intersect.
If you pick and , you have the other two sides of the quadrilateral, which do not intersect.
If you pick and , you have the diagonals of the quadrilateral, which do intersect.
Any four points on the original circle of can be connected to form such a convex quarilateral , and only placing and as one of the diagonals of the figure will form intersecting chords. Thus, the answer is , which is option .
Notice that is irrelevant to the solution of the problem; in fact, you may pick points from the entire circumference of the circle.