One stick, two random cuts, three pieces. A simple question to ask whose answer usually surprises.
The broken stick
Riddle statement
You break a stick at two points chosen at random, and thus obtain three pieces.
What is the probability that these three pieces can form a triangle?
Show solution
Solution
The condition for three segments to form a triangle is that the longest one be less than the sum of the other two.
As the three pieces add up to exactly the length of the stick, this is equivalent to asking that the longest piece measure less than half the length of the stick.
We represent the two breaking points as coordinates within a unitary square of possibilities. Within that square, the favorable region is the one in which none of the three pieces exceeds half of the suit.
When drawing this region, a central triangle results whose area is exactly a quarter of the total area.
Therefore, the desired probability is \(\tfrac{1}{4}\).