Consider quadratic equations $Ax^2 + Bx + C = 0,$ in which $A, B,$ and $C$ are independently distributed $Unif(0,1).$ What is the probability that roots of such an equation are real? This problem is from Chapter 3 of Rice: Mathematical Statistics and Data Analysis (editiona 1 through 3). Until recent printings of 3e, the incorrect answer 1/9 was given for this problem.
However, Horton (2015) http://www3.amherst.edu/~nhorton/precursors/precursors.pdf points out that the correct answer is slightly above 1/4, as can be verified by a simple simulation. (Horton and his colleagues are concerned with elements of an undergraduate curriculum to prepare students in the mathematical sciences to cope with modern data science.)
In a somewhat more practical setting, one might consider a discrete version of this problem. A program that produces random drill problems on quadratic equations $Ax^2 + Bx + C = 0,$ selects values for $A, B,$ and $C$ at random and independently from among the ten equally likely values $0.1, 0.2, \dots, 1.0.$ What proportion of such equations have real roots? And what proportion have only one root?
The initial Answer sketches the exact analytic solution of the original problem and shows numerical and graphical results from simulation. A simulated result for the discrete version is also shown.
Additional answers using other methods or discussing related topics are welcome.