# Collections of nonstandard dice with standard distribution of sums

Jelka Županec (2018) Collections of nonstandard dice with standard distribution of sums. MSc thesis.

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## Abstract

Understanding probability is very important, not only within the fields of mathematics and computer science, but also in everyday life. The final thesis considers collections of nonstandard dice with standard distribution sum. The term “non-standardized” refers to dice with non-standard signs, where the chance of falling on each of the six sides is equally probable. The term “standard (fair) dice” refers to dice with 1-6 dots where the chance of landing on each side is equally probable consider a pair of six-sided non-standard (Sicherman) dice, which have been discussed by different authors (Gardner, Fowler, Swift). These dice are labelled with numbers [1,2,2,3,3,4] and [1,3,4,5,6,8], makingthe same sum as a pair of standard dice. The simplification of the problem of dividing the sum in nonstandard dice to dice with more sides and more dice with n signs is introduced through the notions of probability and algebra, such as probability random variable, mass and generating function and cyclotomic polynomial. The middle part of the thesis generalizes Sicherman's result into pairs and groups of the n-sided dice and seeks the solutions for non-standard pairs of dice in the shape of platonic bodies. It also shows how standard distribution of the sum of dots can be reached with combining dice with different number of sides. Number of solutions of Sicherman's problem for the pair of n-sided dice in the p and pq sizes (p and q not necessarily being different prime numbers) is described in detail. From the research and the collected data interesting consequences for the examples n = 2p and n = p2 are derived. Theoretical results are complemented with computer research. We completed theoretical results with computer research. The last part of the thesis shows some potential uses of nonstandard dice in Maths lessons in primary schools. Worksheets were given to 9th grade students and the level of difficulty of sample worksheets was evaluated. The concept of probability distribution is not included in primary nor secondary school curriculums. It can, nevertheless, be intuitively understood through the introduction of simple examples.

Item Type: Thesis (MSc thesis)
Keywords: probability, random variable, generating function, cyclotomic polynomial, nonstandard dice, probability distribution
Number of Pages: 74
Language of Content: Slovenian
Mentor / Comentors:
Mentor / ComentorsIDFunction
doc. dr. Boštjan KuzmanMentor
Institution: University of Ljubljana
Department: Faculty of Education
Item ID: 5457
Date Deposited: 09 Nov 2018 11:18