EXERCISE IN PROBABILITY NO. II



In the previous paper, we looked at the number of permutations of 3 dice that add up to any given sum. This problem can have valuable applications if it can be extended to n dice where each die has values in the range a to m. We first consider the case where a=0.

The formula was obtained in an empirical fashion. Formulas for 2 and 3 dice were tried first. These formulas were easy to establish as seen in paper I. Extending the formula to more than 3 dice involved counting points in hyperspace. Since that can not be visualized, pattern extensions were used for higher dimensions. The formula has not been verified for very high dimensions because the computation time is too much for a brute force confirmation. However, curves can be plotted and they follow very closely to the shape of a bell curve.

During the pattern extension process, note that a sequence such as 1 and then 2 may be followed by 6 when multiplications are involved. This occurs if the sequence is factorials. Alternating signs may not be apparent with only 2 example formulas. Finally watch for powers of 2 or binomial expansions. When you only see the beginning of a sequence, you may miss the pattern. The reality check comes when you test your formula against the brute force computer program. If your formula remains true for several more terms in the sequence, you are pretty safe but there is no guarantee. Always claim that your formula is hypothetical and it will be acceptable.





We conclude this paper with a file which can be downloaded that computes the distribution curve for n=28 and m=28. It is compared to a superimposed bell curve. Note that the bell curve equation must be adjusted so it peaks at p([(nm)/2]) and the total area under the curve is (m+1)^n. Remember that the normalized curve as described in the first paper has a total area of 1. To help compute the curve, note that the standard deviation should be ((m+1)^n)/(p([(nm)/2])*sqrt(2*pi)). Also remember the curve must be centered at [(nm)/2] where [] implies integer division.



The download file contains Turbo C source code and a DOS executable. If you download and execute it, you will see that the formula in this paper matches a true bell curve almost exactly. It is assumed that the difference is due to the fact that n and m are finite. The main payoff in this paper is the ability to compute p([(nm)/2]). Knowing the peak value of the bell curve, where it is centered, and knowing its standard deviation as described with the above formulas allow you to quickly plot it.



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