The Binomial Distribution: Historical Origin and Evolution of Its Problem Situations

The increase in available probabilistic information and its usefulness for understanding the world has made it necessary to promote probabilistic literate citizens. For this, the binomial distribution is fundamental as one of the most important distributions for understanding random phenomena and effective decision making, and as a facilitator for the understanding of mathematical and probabilistic notions such as the normal distribution. However, to understand it effectively, it is necessary to consider how it has developed throughout history, that is, the components that gave it the form and meaning that we know today. To address this perspective, we identify the problem situations that gave origin to the binomial distribution, the operational and discursive practices developed to find solutions, and the conflicts that caused a leap in mathematical and probability heuristics, culminating in what is now known as the binomial distribution formula. As a result, we present five historical links to the binomial phenomenon where problem situations of increasing complexity were addressed: a case study using informal means (such as direct counting), the formalization of numerical patterns and constructs related to counting cases, specific probability calculus, the study and modeling of probability in variable or complex phenomena, and the use of the distribution formula as a tool to approaching notions such as the normal distribution. The periods and situations identified correspond to a required step in the design of binomial distribution learning from a historical epistemological perspective and when solving conflicts.