About Mathematics Commentary
Chapter 7. Mathematical Models
What this chapter is aboutThe central driving force of this chapter is random number. It is the application of such numbers that gives us the power
to model real world situations for which we have some mathematical rationale. As the remainder of the chapter makes clear,
situations involving probability represent perfect settings for the application of random numbers
A few students may be put off by the preponderance of examples taken from the world of gambling. There are, it seems to me,
two quite reasonable responses to this: (1) they are used because they are simple and thus illustrate the concepts, and
(2) several of them, for example Panel 7.4.2, offer good reasons not to gamble.
Section Notes and Activities:Section 7.1 introduces random numbers and demonstrates how they can be used to model real world activities. The example of modeling the
toss of a pair of dice is important because it illustrates the care that needs to be used with this powerful tool.
While exercise (7.1.2) shows that the generation method fails for some numbers, there is an even more serious problem with
the method: If you know the method, given one value, you know what will be the next value. That means that the numbers are not
truly random. In fact, unless the source of the numbers is some physical phenomenon like a radiation count, any computer-based
random number must have a “method” driving it. That is why such numbers are really pseudo-random numbers. This is worth discussing
with your class, but the problem does not compromise our use of these numbers.
You can demonstrate the problem associated with pseudo-randomness with two programmable calculators produced by the same
company (for example two TI-84s) that have never before produced any random numbers. The first time a random number is called for
on the two calculators, they will be the same, as will be the following series of numbers.
Scientists are always on the lookout for ways to generate truly random numbers. You may wish to call your students’ attention
to a recent article in the journal Matter by Edward J. Lee, et al., “A Crystallization Robot for Generating True Random Numbers
Based on Stochastic Chemical Processes”, which may be accessed at www.cell.com/matter/pdf/S2590-2385(20)30024-2.pdf. The article
will give them a sense of how complex the process is just to obtain what in this case is simply a series of binary digits.
Section 7.2 demonstrates how what we think are random choices lack the features of true randomness. Thus, it illustrates the usefulness
of the random numbers generated by computer, despite the problems discussed above.
You might be concerned about exercise (7.2.3) because nothing is said about how the so-called random triangles are chosen.
Here they are chosen simply by the number of degrees in one angle. Clearly, that choice leads to approximately half of each type.
This is a reasonable concern and you might discuss other ways to think about a random triangle. For example, one way would be to
use the first digit in a series of three random numbers to represent the three side lengths of a triangle and to see if the result
produces an acute triangle, an obtuse triangle or no triangle at all. You could do this as a class project, but you would need
rulers and compasses to address it systematically.
Section 7.3 is a simple (statisticians would probably say simplistic) introduction to some of the basic ideas of probability: in particular
independence.
Exercise (7.3.9) is, in part at least, an example of what has come to be called the hot hand fallacy or more generally the
gambler’s fallacy. An interesting article about these fallacies and some extensions from them is The gambler’s fallacy fallacy
(fallacy) by Marko Kovic and Silje Kristiansen. It may be downloaded from the web.
Section 7.4 about expectation leads to the powerful example of lottery payoffs. Here is an example of the small return over time that the
highly advertised near-national lottery provides its players. In this regard exercise 7.4.4 (b) is very important, because it shows
how the huge payoff, which happens so rarely, affects normal play
Section 7.5 addresses a situation where, as the text says, “expectation lets you down.” The real point of the section is, however, the way a
mathematician has addressed this problem, recognizing that the game’s infinite feature is not possible in the real world.
Discussion questions: Does this mean that math doesn’t work? Should we throw out expectation since it doesn’t apply to this example?
Section 7.6 on Monte Carlo simulations finally gives students some non-gambling examples where random numbers play a role. The important feature of this
section is the comparison of results derived from the use of random numbers with mathematical conceptualization.
Section 7.7 on modeling π is really a continuation of Section 7.6. One thing that is worth observing with your students is how slowly the process converges
on the value we already know.