Sets make up the world. Everything can be thought of as a set, and even the relationships between different things are comprised of them. This abstract and sometimes hard-to-define concept is the beating heart behind most of the field of mathematics. Geometry examines sets of points, algebra examines sets of numbers, and calculus examines sets of functions (which are themselves sets of points). The focus of this paper lies in the connections between probability and set theory. Probability deals with the likelihood of events occurring, and forms the foundation for much of statistics, actuarial math, and other related fields. At the heart of it all lies sets.
Unlike most other fields of mathematics which deal with absolute and complete certainty (such as the exact relationships between sides of a triangle or the precise solutions to an equation), probability deals with the daunting and often incomprehensible nature of uncertainty. Perhaps due to this, advances in probability have only occurred fairly recently. Throughout the history of the world, people have always been fascinated by games of chance and uncertainty, mostly due to the exhilaration of unlikely events happening. However, it was not until the fifteenth century that this began to be formalized, and the 1713 book Ars Conjectandi (the Art of Prediction) by Jacob Bernoulli cemented this as a legitimate area of study (Debnath and Basu, 2014). In contrast to this, there is evidence that even as far back as 1650 B.C.E. of people in Egypt solving equations for one unknown, and algebra was solidified closer to 825 C.E. (Corry, 2023). Thus, probability is a relatively new and innovative field.
Set theory's development also finds itself very recent. Although so many mathematical concepts and even applicable concepts are seen today through the eyes of sets, the formal structure and legitimization of set theory did not come until Georg Cantor's work in the turn of the 20th century (Bagaria, 2023). Cantor wrote extensively of different "sizes" of infinity and developed formal definitions for cardinality. This was the first and foundational treatment of formal set theory, though it did lead to many contradictions and paradoxes, many of which were addressed by the axiomatization of set theory by Zermelo (Bagaria, 2023). Thus, a critical piece of mathematics has only been expanded on and reasoned with for less than 200 years. Therefore, there is much effort to be done in the understanding and distillation of these topics. Thus, the focus of this paper is in a deeper exploration of these topics.
With an understanding of the history behind these concepts, we will now examine these concepts and more particularly their relationships in greater detail. First, we will look at set theory and the ideas behind it. As has been stated, the concept of a set is very hard to nail down. The Encyclopedia Britannica notes that it "deals with the properties of well-defined collections of objects" and says that a set "is regarded as being a single object" (Enderton and Stoll). Additionally, the order and multiplicity of items in a set is insignificant to their behavior. Thus, sets can be thought of in regards to many different circumstances and can contain a wide variety of objects. An individual set may not be particularly interesting, but the relationships between them are. One of the most common is the concept of a subset, where each element of a subset is in the original set. There are also operations defined between sets, and some common operations on two sets include the intersection which produces the set containing all elements that are in both, the union which produces a set with elements in either one set or the other or both, and the Cartesian Product (notated with x), which produces a set of all ordered pairs (a, b) where each a is in the first set and each b is in the other set (Sundstrom, 2022). Additionally, all sets can be defined in a particular universe, which is itself another set (and may even contain sets). The complement of a set is defined to be the set made up of all elements of the universe which are not in that set.
Probability examines the likelihood of an event occurring. Looking at a particular set of outcomes to an experiment (any process producing outcomes), probability seeks to compute the relative likelihood of each of those outcomes (or sets of outcomes, as we shall see later). This can best be thought of as the percentage of time that an event would occur in an infinite number of repetitions of that experiment, or if all outcomes are equally likely (as seen in the applet below), it can be thought of as the ratio of the number of outcomes satisfying a condition we set to the total number of outcomes (Khan Academy). Therefore, the probability of an event can be expressed as a number between 0 and 1, where a probability of 0 means that it never occurs and a probability of 1 means it will always occur, with something in between indicating it sometimes happens. Another way of thinking this is as a function (which is a set of ordered pairs) in which the domain is the set of events and the codomain is the interval [0,1]. Thus, each event is mapped to a singular probability. The details of this function lies at the essence of probability. We must think about combinatorial viewpoints to understand what makes up the events, and we often need to use calculus to add continuous variables. This is a very nuanced and intricate process.
In regards to the connection between probability theory and set theory, the most direct conclusions lie in relating the outcomes to sets themselves. Each time we look at a probability question, we need to think of all possible outcomes of an experiment as a set (which is often called the sample space). That is, it is a singular entity we can examine. The elements of that set will be outcomes, which are specific happenings or sequences of occurrences. The beautiful part about set theory is that we do not necessarily need to know what those outcomes actually mean. However, once the set of outcomes is made, events must be defined. These are subsets of the sample space (so they are themselves sets) which we are interested in, which can be simple (consisting of one outcome) or compound (BYJUS, 2023). Along with this, we can think of relationships between events according to the set operations between them. That is, we can make potentially compound events by defining a new event as the union or intersection of two other events (colloquially thought of as one event or the other happening or both). When looking at conditional probabilities, we examine the behavior "when the evidence or information that is given is more specific than what is captured by initial set of outcomes" (Hajek, 2011). These are one of the most critical and complex questions of statistics, but one most infrequently related to set theory. We can think of conditional probabilities as restricting the sample space based on some condition, so the domain of the probability function has changed from all events to only those based on the condition. Thus, each time that a condition is added to a probability, the function must be redefined. This function lies at the heart of probability. A function is a set of ordered pairs, where in this case the first element is the event we are looking at, and the second element is a real number between 0 and 1. The set of these ordered pairs makes up the probability function we look at. Thus, at the end of the day probability is simply a matter of finding what the event is we are looking at and determining the definition of the function such that this maps to the correct value.
Probability is perhaps one of the most misunderstood and daunting topics for students to face. A study from 2018 found that university students in the statistics class they surveyed did not have "pleasant" views about probability and lack confidence in solving problems relating to it, despite seeing the importance of the topic (de Oliveira et al, 2018). Thus, finding ways to help them overcome this fear and obstacles would be extremely beneficial. Set theory is not often explicitly taught but rather is a hidden figure behind other mathematics. Thus, giving students a clear focus on set theory (perhaps even independently of probability) allows them to better understand relationships between events and probability functions, giving a deeper lens to why probability is built and examined the way it is. Mathematics is all about understanding problems from a more abstract point of view, and set theory allows the learner to understand probability from that perspective. Thus, this relationship that this paper has examined allows all to comprehend probability from a new and objective viewpoint.
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Khan Academy. (n.d.). Probability: The basics (article). Khan Academy. https://www.khanacademy.org/math/statistics-probability/probability-library/basic-theoretical-probability/a/probability-the-basics
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de Oliveira Junior, A. P., Zamora, P. R., de Oliveira, L. A., & de Souza, T. C. (2018). Student's Attitudes Towards Probability and Statistics and Academic Achievement on Higher Education. Acta Didactica Napocensia, 11(2).