Quick counting of probability distributions for the sum of multi-sided dice: a physical and didactic approach

There are many physical systems (with randomness) where the performance of the probability distribution is of great importance [for example the so-called radioactive dice (Murray and Hart 2012 Phys. Educ. 47 197-201)]. A relevant situation is when the value (positive integers) of the upper faces is added when rolling fair dice (unbiased), that is, balanced, regular or unshaven. It has been widely studied in the basic literature for one, two and a maximum of four dice. However, a generalization using these methods for many dice is too cumbersome to carry out. In this paper we explore a method that uses the moment generating function, already studied in Singh et al (2011 UNLV Gaming Res. Rev. J.), for five six-sided fair dice with positive integer values. We carry out the generalization of this method to study the performance of the probability distribution and make a quick counting without complications for the sum of n fair dice with s sides (s > 6), and real values. We also study the invariance of the distribution when we set the number of sides, that is, for s-fix. Therefore, we study the distributions for the outgoing total sum (random variable) of roll generalized dice, for combinations of n multi-sided dice.