The Marble Bag Experiment We have used a rather unconventional probability experiment in our physical chemistry laboratories for the last few years. This experiment was developed after we became convinced that our students could only work with probabilities in one direction. By this we mean that they were reasonably adept a t going from a given situation to the probabilities whieh would be expected to fallow from it. However, they had little or no experience in beginning with the results and working back t o the situation from which they arose. In its most usual form, the students (usually a team of two) are given a bag which contains marbles of four different colors and are told that the total number of marbles in the bag is a prime number. They are to find both the distribution and the total by drawing (and replacing after each draw) the marbles from the bag, two a t a time. How many pairs they draw is left entirely up to them. Usually an hour's worth of data collection is sufficient. A faster, rough (on the marbles), and noisier method for collecting data employs a dishpan instead of the bag. We are sure that one or two rapid agitations of the dishpan ensures randomness. A variation we have often employod is that the students, when ready to do so (again their decision), may request a "wild marble" which they add t o the bag. If i t appears likely that the students will have trouble analyzing their data, this marble can be of a new, fifth color. However, we have usually let the new marble he the same color as one of the originals, a tactic which forces the students t o think more. We do not tell the students how to analyze the data hut suggest that they devise their awn scheme. Obviously many equally acceptable schemes are possible. Rather than spoiling the fun by delineating some of these, we prefer only t o point out the most common failing our students exhibit. With four marble colors, ten different possible pairs may be drawn. Because i t is more difficult t o include the data from the two color pairs, some students decide t o omit this data. This is, of course, wasteful of their tediously collected data and we inform them of this. It also, especially in the case of a n unabundant marble, puts too heavy reliance on the results for a pair whieh was drawn only a few times. While we cannot make any precise analogy between this experiment and any one physical chemical measurement, we feel it illustrative of a situation which often oecurs. One is presented with the "answer" without full knowledge of the "question" which led to that "answer". This can be shown a t two levels. In atomic spectroscopy one is given a set of energy differences and must find the set of energy levels into which they will fit. In normal coordinate analysis in vibrational spectroscopy one has a set of frequencies (the "answers") and must find the set, or a set, of force constants (the "questions") whieh would lead to those "answers". Williard L. C r a f t Adrian College Adrian. Michigan 49221
David L. Powell College of Wooster Wooster. Ohio 44691
