There are no answers in the back of the book of life. Our former students need to be able to validate their own mathematics. The collection of problems presented here include solutions based on random numbers plus one or more calculus solutions. Perhaps the best way to gain confidence that an answer is correct is to work the problem in at least two independent ways and obtain consistent answers.
These problems each look at the average value of some geometric object. They are easy to state, and students with access to a system with a random number generator can produce reasonable approximations for these average values. An understanding of these problems will not only give students a variety of techniques for solving such problems, but will also provide an intuitive feel for additional study in areas such as probability, statistics, and mathematical expectation.
Although most of the integrals are straightforward to write down, finding closed-form solutions range from easy to extremely challenging. Indeed, many of the closed-form solutions referred to here have been given for the first time within the past few years, some within the past decade. See Weisstein, Eric W. “Ball Line Picking.” From MathWorld–A Wolfram Web Resource. http:// mathworld. wolfram. com/ Ball Line Picking.html and related MathWorld references for details on such problems.