The Broken Spaghetti Noodle

A popular question in recreational mathematics is the following: If we drop a spaghetti noodle and it breaks at two random places, what is the probability that we can form a triangle with the three resulting segments? See for example [2, chap.1, sec.6], [3, p.6], [4, p.31], or [7, p.30-36]. This is an elementary problem in geometric probability. Clearly the length of the noodle (or equivalently our choice of unit length) does not matter, so the problem amounts to choosing two numbers at random from the interval (0, 1), say a and b with a < b, and looking at the resulting intervals (0, a), (a, b), and (b, 1). We will be able to form a triangle when the positive numbers a, b−a, and 1− b satisfy the triangle inequality (i.e., when no interval is longer than the combined lengths of the other two). Equivalently, this will be the case when all three intervals have length less than 1/2. Therefore, a triangle can be formed precisely when the following three inequalities hold: a < 1/2, b− a < 1/2, and b > 1/2. Figure 1 shows all possible outcomes 0 < a < b < 1, and the darker shaded region consists of all “favorable” outcomes, when a triangle can be formed. Comparing areas, we see that the probability of succeding in getting a triangle is 1/4.