Nabori nestandardnih kock s standardno porazdeljeno vsoto pik : magistrsko delo

Županec, Jelka (author), Kuzman, Boštjan (mentor)

Razumevanje verjetnosti je pomembno področje ne le v matematiki in računalništvu, ampak tudi v vsakdanjem življenju. V magistrskem delu obravnavamo nabore nestandardnih kock s standardno porazdeljeno vsoto pik. V tem delu kot nestandardne kocke razumemo predvsem kocke z nestandardnimi oznakami, kjer so vse ploskve enako verjetne kot pri standardnih (poštenih) kockah, ki so označene z oznakami od 1 do 6 in je vsaka ploskev enako verjetna. Za izhodišče vzamemo par šeststranih nestandardnih (Sichermanovih) kock, o katerih so pisali različni avtorji (Gardner, Fowler, Swift). Omenjen par nestandardnih kock je oštevilčen z oznakami [1, 2, 2, 3, 3, 4] in [1, 3, 4, 5, 6, 8] in ima enako porazdeljeno vsoto pik kot par standardnih kock. Za obravnavo posplošitve problema porazdelitve vsote pik na parih nestandardnih kock na kocke z večjim številom ploskev in na več kock vpeljemo v uvodu pojme iz verjetnosti in algebre, kot so verjetnost slučajne spremenljivke, njihove porazdelitvene in rodovne funkcije, ter ciklotomični polinomi. V osrednjem delu naloge Sichermanov rezultat posplošimo na pare in trojice n-stranih kock in poiščemo vse rešitve za nestandardne pare kock v obliki platonskih teles, torej tetraedra, oktaedra, dodekaedra in ikozaedra. Pokažemo tudi, da lahko standardno porazdelitev vsote pik dosežemo s kombiniranjem kock, ki imajo različna števila ploskev. Natančno opišemo tudi število rešitev Sichermanovega problema za par n-stranih kock velikosti p, pq, kjer p in q nista nujno različni praštevili. Od tod izpeljemo tudi zanimive posledice za primera n = 2p in n = p2. Teoretične rezultate dopolnimo tudi z računalniškim preiskovanjem. V zadnjem delu magistrske naloge je prikazanih nekaj možnosti za uporabo nestandardnih kock pri pouku matematike v osnovni šoli. Pripravljena delovna lista »nenavadne kocke« smo preizkusili tudi na učencih devetega razreda in ovrednotili zahtevnost obravnavanih vsebin zanje. Koncept verjetnostne porazdelitve v učnem načrtu za osnovno šolo niti za srednjo šolo sicer neposredno ni zajet, vendar ga lahko osnovnošolci vsaj intuitivno razumejo preko primerno predstavljenih enostavnih zgledov.

• Language: Slovenian
• Keywords: slučajna spremenljivka, rodovne funkcije, ciklotomični polinomi, nestandardne kocke, verjetnostna porazdelitev
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: PEF - Faculty of Education
• Publisher: [J. Županec]
• Year: 2018
• Number of pages: 74 str.
• PID: 20.500.12556/RUL-105337
• UDC: 510.643.7(043.2)
• COBISS.SI-ID: 12202569
• Publication date in RUL: 10.12.2018

Secondary language

• Language: English
• Title: Collections of nonstandard dice with standard distribution of sums

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.

• Keywords: probability, verjetnost