Quick counting of probability distributions for the sum of <i>biased multi-sided dice II: Cluster Bias</i>

Abstract In a previous paper (2022 Eur. J. Phys. 43 045801), we carried out a study on quickly calculating the probability distributions of many dice with a number of faces greater than 6 (it can also be less). This formulation allows us to unambiguously obtain the graphical and tabular distribution of the random variable of interest—that is, X = ∑ j X j —where we were interested in the total sum of the resulting sides of the dice. We also involve the concept of radioactive dice, giving a more physical and didactic approach. We generalize the moment–generating function (MGF) method to include real and imaginary numbers. Now, in this complementary work we research the situation in which an unconventional dice system has a grouped or clustered bias; that is, arbitrarily choosing the probabilistic weight assigned to each side of the dice, which can happen absolutely randomly or be manipulated. We show the mathematical characteristics—statistical–probabilistic—that must be fulfilled, as well as examples with simulations for different cases, which illustrate the formulation effectiveness. We introduce the concept of cluster bias and the cluster bias probability distribution function (additive cluster bias and multiplicative inverse cluster bias). Also, we give a brief perspective of applications in other scientific branches.