lew (02/26/83)
The following was inspired by Jan Wolter's request for information about irregularly shaped dice. I carried out some probability experiments using loaded dice. These were standard dice, modified by drilling and loading. I have labelled the three dice involved as "light", "heavy" and "hole". The light die was a die with the "one" side drilled out. The heavy die was the same die with the hole filled in with lead shot and epoxy. The results of the trials using these dice suggested to me that the inertia tensor was important to the probability distribution, so I tried drilling a hole straight through another die. The idea was that the CM should still be in the center but the inertia tensor would break the cubic symmetry. The results with this die corroborate my hypothsesis based on the first two, namely that a prolate inertia tensor will "prefer" to orient vertically, and an oblate inertia tensor will "prefer" to orient horizontally. I fitted the data for each die to what I call the static model. This model predicts the probability of falling on each face to be proportional to the area of the face projected onto the unit sphere with the CM at the center. This gives a 3-parameter fit to data with five degrees of freedom, leaving two degrees of freedom for the chi-squared fit. I will not go into the justification for using the chi-squared fit in this experiment, but I am prepared to defend it. The three parameters are the CM coordinates. The geometric center of each die is (.5,.5,.5) John Aspinall suggested this "static model". I proposed a "kinematic model" which is based on the same unit sphere. The results from the "light" die suggested to me that my kinematic model might give a better fit, but the results from the "heavy" die deviated in the opposite direction from what I qualitatively expect from the kinematic model. I expect the kinematic model to favor the heavy side at the expense of the four "neutral" sides. As a result, I didn't evaluate the kinematic model but developed my inertia tensor idea. The third,or "hole" die does support this idea, but still with an 8.2% probability of rejecting the null hypothesis (correctness of the static model.) I could go on at great length about various aspects of the theory and experiment but I will just present the data without further ado, except to point out that the "1" and "6" faces are fitted low in the case of the light and hole dice (prolate inertia tensor), and fitted high in the case of the heavy die (oblate inertia tensor.) light heavy hole CM .369 .508 .520 .691 .463 .468 .493 .490 .501 chi2 17.88 5.27 4.97 Prej0 .0001 .071 .082 trials 10094 2945 5707 face fit obs fit obs fit obs 1 2261.0 2330 327.8 294 965.8 1005 6 1271.5 1381 756.0 740 936.5 976 2 1610.4 1590 503.1 512 972.4 950 5 1667.9 1647 428.2 438 930.4 905 3 1569.6 1496 497.3 512 948.9 933 4 1713.6 1650 432.6 449 953.1 938 NOT INSANE! Lew Mammel, Jr. ihuxr!lew