The Empirical Foundations of Probability Theory.

IT IS desirable that scientifically ori ented secondary and college students be exposed to the role played by mathemat ics in modeling physical theory and ex periment. The nature of the probability model and the empirical evidence offered to support its applications can furnish a very effective counterpoint to the stan dard examples of the empirical evidence from astronomy and physics that justifies theoretical mechanics. Mathematical physics finds its empir ical justification in the ability of the phys icist to calculate from the initial state of a system its physical evolution. Initial data are collected for a designed or nat ural experiment, the calculations are made, the experiment is conducted, and a termi nal state is measured. If the physical theory behind the experiments is correct, the measurements of the terminal state agree with the values predicted by calcu lation within experimental error, and the theory is "proved." Most good students are aware of this scientific method and know that observational astronomy is the great classical example of its application. Some students will be aware that mathe matical theories are approximate models of reality and that as measurement im proves a new theory may be necessary to explain discrepancies in measurements that were ascribed to experimental error in previous experiments. The great his torical examples of this process?the phys ical revolutions of Kepler, Newton, and Einstein?should be familiar to every sci entifically oriented student. A student of elementary probability who has an understanding of these stan dard examples of the scientific method is ready to consider the empirical evidence that "proves" probability theory. The in structor may ask the student to consider a simple model such as the thirty-eight equally likely outcomes for one spin of a standard Las Vegas roulette wheel. The student can be asked if the following ex periment is a satisfactory way of proving that this model is correct. The instructor will predict that the ball will not land in green double zero ; he will spin the wheel, and if the ball does not land in double zero, he will have proved that his prob ability model for the roulette wheel is cor rect. The student will promptly say that this is not acceptable. It will shock the student to discover that all the standard examples that prove probability theory are no deeper and no better than this ex ample that he is so quick to label unac ceptable. The student will say that you cannot use just one spin of the wheel to prove your model?you need many spins, say 10,000. The instructor should then point out that the probability model for 10,000 spins is a probability model with 3810'000 outcomes and that the student is now trying to prove that this model is correct. How does the student propose to justify this model? The student will say that each number should appear with relative frequency about 1/38 ?this would prove the model was correct. After further discussion the student will realize that he must state before perform ing the experiment what outcomes will prove the theory and that the set of these outcomes is an event in the probability space. If the theory is correct and the stu dent expects to demonstrate it in this