Probability that a cubic has two turning points

Most cubic polynomials with real coefficients have two turning points, a local maximum and a local minimum. But how do you quantify most"?

Here's how one author did it [1]. Start with the cubic polynomial

x^3 + ax^2 + bx + c

Since multiplying a polynomial by a nonzero constant doesn't change how many turning points it has, we might as well assume the leading coefficient is 1.

In his paper, Robert Fakler assumes a, b, and c are chosen randomly from an interval [-k, k]. He shows that for k 3, the probability that the polynomial has two turning points is

p = (9 + k)/18.

For k >= 3, the probability is

p = 1 - (3/k) / 3

and so as k , p 1.

[1] Robert Fakler. Do Most Cubic Graphs Have Two Turning Points? The College Mathematics Journal, Vol. 30, No. 5 (Nov., 1999), pp. 367-369