Probability and Combinatorics

Combinatorial calculus is a branch of mathematics oriented to determine the number of distinct cases that can occur in an experiment or the number of elements that make up a set. One question often remains open in the debate about the role of combinatorics: Does the introduction of combinatorics in school teaching have an autonomous mathematical value or is it a simple tool for teaching probability? Problems of a combinatory nature can be found in many areas of mathematics, but in this chapter, emphasis will be placed on a limited circle of topics with a close link to probability. The objects presented will serve as an introduction to situations that technically relate to permutations and combinations.

Frequentist Probability

Frequentist probability is historically presented as an attempt both to overcome the limitations of classical conception and to take into account the impressive development of experimental sciences and statistics. It is precisely because of this close link with the statistical sciences that it finds a significant place in teaching. It is also an area in which the use of IT tools is crucial. This approach also makes it possible to calculate the probability of events and from it we derive the same rules examined in the classical definition, since it is sufficient to replace the ratio between the number of favorable cases and the number of possible cases with the limit of the ratio when the repeated tests tend to infinity. One of the fundamental concepts appears in the chapter, that of a random variable capable of describing events and their distribution.

Classical Probability

Classical probability is historically presented as the first interpretation of random phenomena. Even with its application limits, it represents the starting point for the main concepts of probability calculation. In class it is the most represented formulation, and, in some cases, it is presented as a standard definition of probability. In the chapter, after presenting the classic objects (coins, marbles, dice, playing cards) normally used to introduce the theme, the panorama is widened thanks to the introduction of less common artifacts and the use of examples that can provide useful insights for further study. The path starts from the concepts of trial and event and ends with the Bayes theorem.

Probability and Simulations

This chapter is fundamental in educational terms as it deals with a topic that cannot be underestimated: the use of laboratory simulation. As long as we talk about probability in the classic sense, paper, pen, and maybe a pocket calculator are sufficient tools, but when we want to analyze a probabilistic model in depth, the use of a computer tool is essential. This support not only allows us to confirm hypotheses but, in some cases, it is indispensable. In the school it is unthinkable to work on large data sets (often not available) that would require not only appropriate software but also a different approach. The balance should be sufficiently significant but easy to handle dataset.

Probability and Statistics

Statistics and probability both fall within the broader scope of the theory of random phenomena. The first deals with providing probability distributions adaptable to the various real random phenomena, and the second deals very often with a random sample to describe its properties or infer to the underlying probabilistic model and the estimation of its parameters. The chapter tries to show this connection by reporting examples that are more or less known but that are characterized by being unconventional. Other objects could have been taken into consideration but those chosen are characterized by a closer link with the calculation of probabilities.

Probability and Distribution

The concept of probability distribution is one of the most relevant but, in some ways, among the most complex concepts to be treated at school. If we exclude some fairly simple cases, there is a variety of distributions whose formulation and use is not trivial. In this chapter, many distributions have been omitted (e.g., exponential, t student, Weibull, etc.) because they are too specialized and reserved for a higher level of study. The choice has privileged distributions that can be analyzed starting from objects with which students are, or maybe, familiar. However, the distributions addressed are sufficient to give a fairly broad idea of the topic.

Probability and Game

Probability is generally concerned with dealing with problems of a random or uncertain nature. The fact that it arises and develops from the analysis of gambling is something that cannot be overlooked. From the point of view of teaching, in addition to historical aspects, it is important to point out the importance of putting students in front of situations that, if not known, can lead to incorrect behavior and pathological attitudes. For this reason, the authors tried to emphasize not only the theoretical aspects, but above all the certainty that you always play “against the dealer” with an expected loss assessable for the various games.

Subjective Probability

Subjective probability aims to solve a big problem: how to calculate probability when none of the definitions examined in the previous chapters is applicable. The probability that it will rain tomorrow, that a newly graduated student will find a job, or that a basketball player will pass the basket record are examples of the subjective conception of probability, as they require neither the knowledge of the mechanism that regulates the phenomenon nor the repeatability of the phenomenon. The value attributed depends on the state of information of the person making the assessment. The already mentioned Bayes' Theorem shows how to make probability evaluations and subsequently modify them in the presence of new data. Faced with the vast field of application, there is a very scarce (one would have to say nothing) attention on the part of the teachers who prefer to concentrate the little time dedicated to the subject to more standardized situations.

Probability and Chaos

Chaos theory is a relatively recent and often misunderstood field of study because it is wrongly considered by laymen as tied to random events. Chaotic systems have properties similar in many ways to those of stochastic processes, for example punctual unpredictability. What is interesting is the fact that potentially controllable situations such as deterministic ones are instead difficult to interpret. It is quite natural to consider some of these situations in a text that deals with probabilities. Even though we continue to stress, these are different fields. To make their way through the many possible examples, fairly simple objects have been preferred.

Probability and Strategies

When you participate in a game, but in general when you compete with one or more opponents, it is essential to define strategic behavior. Because probability often plays a central role, it is important to learn how to profitably manage the information we have. The chapter presents a multiplicity of games, more or less known. The whole allows us to take up some already known concepts and to suggest new ways of deepening them. It should be noted that the center of the analysis is on probability and we do not want to focus only on games of chance, even if they constitute a field historically very studied.