This article explores three classical problems in probability: the birthday problem, the poker problem, and Buffon's needle problem. The birthday problem is famous because the event of interest has much higher probability than one might first expect. With poker, the situation is just the opposite; many of the desired events (unfortunately for the gambler) are very rare. Buffon's needle problem is surprising in its very concept, has great historical interest, and illustrates the beautiful interplay between probability and statistics. The birthday and poker problems come from discrete models; Buffon's needle problem involves a continuous model. Yet the three problems share a couple of key properties: each is mathematically simple, but still leads to results that are interesting and a bit surprising.
The three problems require relatively little in the way of mathematical prerequisites, other than a bit of set theory and a basic understanding of the language and notation of probability:
In the birthday and poker problems, the probabilities of interest are computed using simple counting arguments. In Buffon's needle problem the probability of interest requires one simple integral.
Of course, these problems are treated in just about every elementary probability text, so what makes this article different? Well, the main important feature is the use of interactive Java applets to illustrate the random experiments associated with each of the three problems. Applets are particularly valuable pedagogical tools in probability. The key concepts and results in probability involve the long-term behavior of random experiments, and so unlike most other areas of elementary mathematics, there are few simple "blackboard sketches" that can be used to illustrate these concepts and results. Moreover, it is usually quite impractical to perform the actual random experiments in a classroom setting. After all, who has the classroom time to deal thousands of poker hands? Applets, by contrast, can simulate thousands of runs of a random experiment in a few seconds and can display the results in a set of coordinated graphs and tables that create a rich learning environment.
Moreover, applets in probability can be constructed that stand alone, apart from any mathematical exposition, as virtual versions of random experiments (interactive micro-worlds). Such applets can be used at a variety of mathematical levels, accompanied by the mathematical exposition that is appropriate for each level.
The applets in this article have a common look and feel. The main toolbar contains Step, Run, Stop, and Reset buttons for controlling the simulation of the random experiment. The Step button performs the experiment one time and then updates the tables and graphs. Additionally, a sound is played that depends on the outcome of the experiment. The Run button runs the experiment repeatedly, with no sounds. The frequency of screen updates can be controlled with the Update Frequency drop down box, and the number of runs before the simulation stops automatically can be controlled with the Stop Frequency drop down box.
For each applet, the first table records the outcome of the experiment (that is, the values of the key random variables) on each update. In addition each applet contains graphs and tables that display special information for that applet. The parameters of the each experiment can be varied with scrollbars.
The applets in this article come from the Probability/Statistics Object Library, an NSF supported (DUE 0089377) virtual library of web-based "objects" for teachers and students in probability and statistics. Generally, the objects in the library are of two basic types:
All objects in the library (including source code) are freely available for use, modification, and re-distribution under the terms of the General Public License.