The Dice Experiment applet (like the other applets in the library) contains no explicit mathematical exposition and thus, in principle, can be used by teachers and students at various levels. The applets in the library are intended to be small "micro-worlds" in which students can run virtual versions of random experiments and play virtual versions of statistical games.
With appropriate exposition, you could use the Dice Experiment applet as part of a discussion of any of the following topics:
- Random experiments and random variables. Rolling dice is a simple and conceptually clear example of a random experiment, and it is one that every student has actually performed. Random variables such as the sum of the dice scores and the largest of the dice scores are also easy to understand, and they are variables that most students have computed playing real dice games.
- A random sample from a distribution. The dice are all governed by the same underlying probability distribution, so rolling n dice generates a random sample of size n from this distribution. The distribution that governs the dice can be specified arbitrarily, so the die in this experiment is really just a simple metaphor for a random measurement that is repeated n times independently.
- The sample mean and the law of large numbers. The law of large numbers shows up in several places: the convergence of the empirical mean and standard deviation to the distribution mean and standard deviation, respectively, and the convergence of the relative frequencies to the corresponding probabilities.
- The central limit theorem. No matter what probability distribution is given to the individual dice (as long as it is not a point mass at a single value), the distribution of the sum and the distribution of the average become more "normal" as the number of dice increases.
- Order statistics. The minimum and maximum scores are the extreme order statistics for the random sample. Their distributions converge, respectively, to point mass at the smallest and largest scores with positive probability.
- Bernoulli trials and the binomial distribution. In terms of rolling an ace or not, the dice form a sequence of Bernoulli trials, and random variable Z has a binomial distribution.
The important point -- and the basic assumption of the PSOL -- is that instructors must provide appropriate expositions of the topics that are suitable for their classes. The Dice Experiment applet and the other applets in the library are of little value without such guidance from instructors. In the language of reusability (see the Reusable Learning project), the applets are adaptable, but not adoptable.
For our discussion, it might be useful to use the term module to refer to a collection of elements (typically including mathlets, exposition, and exercises) that is focused on a relatively small mathematical topic and is pedagogically complete. Most interactive materials on the web, including those in this journal, the MathForum collection, and other portal sites, are modules or even larger learning environments. In many cases, the elements of a module are tightly coupled and cannot be used independently -- such modules were never intended to be broken apart and adapted to other settings. In short they are adoptable but not adaptable. Clearly, a well designed module has two main advantages:
- Very little work is required by the instructor to use the module.
- The module can be sharply focused on particular learning objectives.
The main disadvantage of modules is that instructors must find ones that precisely fit their needs in terms of content, level, and learning objectives. Moreover, combining modules from different sources is likely to result in a confusion of conflicting styles, notation, and user interfaces.
One of my goals for the PSOL is to provide general-purpose applets in probability and statistics that can be used as elements of high quality modules. I believe that both types of resources are useful: complete modules and libraries of reusable elements. However, there are not as many collections of the second type. Moreover, developers must understand the inevitable tradeoffs between the two approaches. The Reusable Learning project has extensive information about reusable learning resources, including guidelines for developers.