Points on a Stick Two points along a straight stick are randomly selected. The stick is then broken at those two points. Find the probability that the three resulting pieces can be arranged to form a triangle. (This is possibly the most difficult exercise in this book.)

Solution 49BB

Two points along a stick is randomly selected. The stick is then broken at these two points. It is needed to find the probability that the three resulting pieces can be arranged to form a triangle.

In any triangle sum of any two sides of the triangle is strictly greater than the third side. Denote the left half of the stick by ‘LH’ and the right half by ‘RH’. If two points from the same half is selected then the sum of the pieces from the same half is less than the other. On the contrary if the points are selected from the different halves then the sum of any two pieces is greater than the third piece (This is intuitively clear). By the abuse of the above notation denote by ‘LH’ when a point is selected from the left half and by ‘RH’ when the point is selected from the right half.