Introduction
Mathematical concepts are ubiquitous in real life, and practical applications of mathematics are not confined to the pages of textbooks or academic tests. Different directions and branches of mathematical knowledge are found in areas that, at first glance, have nothing to do with the need for calculations, equations, and structuring, whether in everyday life, the clinical industry, or any other sector. The multiplicity of applications of mathematics is due to both the versatility and complexity of this science, as well as the fact that mathematics studies the ways and possibilities of order.
This research paper focuses on probability and counting as areas of mathematical knowledge. Both related areas are scrutinized and critically examined in both scientific and practical terms, and described in terms of their relationship to the real world. The importance of such a study stems from the desire to deepen knowledge of selected topics in the mathematical sciences and to provide a framework for proving the breadth of applications of mathematics in the real world.
Brief Description of Selected Concepts
Probability and counting are two closely related areas of mathematical knowledge, branches of statistical study. On the one hand, probability is the degree to which the possibility of a particular event occurring is assessed. For example, in a fair game of coin flipping, a coin has only two sides so that each side can fall out with equal probability. Probability should be defined as the ratio of the number of favorable events (x) to the total number of events (N), as shown in Equation [1]. The more favorable events in the sample, the higher the probability of occurrence.
p=x/N [1]
Several rules must be understood in advance before doing mathematical calculations with probabilities. First, two events are called independent if the outcome of one does not affect the occurrence of the next. For example, if one takes two dice and rolls them (simultaneously or in turn), the number that falls on the first die is not related to the second die’s value — these are independent events. On the contrary, events will be classified as dependent if their outcomes are connected: drawing a specific card from the playing deck one by one depends on the previous attempts because, after each round, the total number of cards decreases.
The probability of two independent events co-occurring (specific values on two dice) will be calculated as the sum of each probability, as shown in Equation [2]. Multiplication of probabilities is used when there is not a set of two outcomes (AND). Still, their intersection (OR) is studied: the probability of occurrence of either value on the dice is calculated as shown in Equation [3]. The previous reasoning is valid for independent events; however, for dependent events, such as those with a notion of conditional probability, the outcome of one event given that the second event has already occurred, as shown in Equation [4].
p(A anb B)=p(A)+p(B) [2]
p(A or B)=p(A)∙p(B) [3]
p(A | B)=(p(A and B))/(p(B)) [4]
On the other hand, a branch of counting in mathematical statistics involves many mathematical operations. One of the simplest and most straightforward frameworks is the mean, which determines the central tendency in a numerical distribution, as shown in Equation [5]. The mean is always related to the standard deviation, which is the amount by which each value in the distribution deviates from the sample mean (Equation [6]). For example, if one takes the age of each student in a class and makes a list, this will represent a numerical distribution of ages: their mean [5] will show the average height in the class, and the standard deviation will report how much, on average, each student’s height differs from the average age of the class.
Equation [7] shows the mathematical definition of variation: strictly speaking, a measure of the spread of data in a distribution. The higher the variance, the more the numbers in the distribution are scattered relative to each other, or the greater the difference in all student ages. The counts section is also closely related to the concepts of combinations and permutations, that is, working with a finite number of elements in discrete distributions.
First, combinations are defined as determining the number of possible outcomes when order is not essential; Equation [8]. If order is important, as shown in Equation [9], it is called a permutation. For a simpler understanding, when one wants to make a pizza topping out of ham, mushrooms, and cheese, it makes no difference in which order the ingredients are taken; combinations deal with this. If, however, it is known that the password consists of 3 of the nine existing digits, then the order turns out to be important — it is a permutation.
Real Life Example: Epidemiology
Epidemiology is one of the critical areas in which the application of the two branches of mathematics described is fundamental. Strictly speaking, epidemiologists study patterns of disease spread in communities and predict outbreaks or declines depending on the factors involved (Brauer et al., 2019). Mathematical probability is used in epidemiology to calculate the probability of disease spread between individuals. For example, during the COVID-19 pandemic, public health agencies discussed strategies to reduce the likelihood of infection, which were implemented based on calculating probabilities for a given virus.
Another focus was investigating the efficacy of vaccines developed, especially when competition between existing drugs began (Andrews et al., 2021). Counting tools also have practical applications: calculating means and standard deviations provides a basis for descriptive studies of disease spread (mean number of people who received immunity or mean number of people who recovered). Combinatorics can be used to estimate potential combinations of encounters between different groups of people, including the most vulnerable. When combined with probability tools, this creates valuable insights into the risks and potential effectiveness of countermeasures.
Both branches of mathematics create a valuable practical tool for solving real public health problems, from which evidence-based recommendations for risk reduction can be derived. Better prediction of disease spread, reduction of adverse outcomes (such as infection and death), creation of countermeasures, and calculation of the effectiveness of drugs and vaccines are the primary benefits of applying probability and counting concepts to public health. Such studies would help communities because the spread of infections would not be controlled by science or linked to real data; strategies would be ineffective or even harmful, and their impact would be reduced. These tools improve the efficiency and accuracy of the interventions developed.
Real Life Example: Economic Trends
The mathematics described above is valuable for financial analysis, particularly investment portfolio management. One of the key parameters of investment performance is the profitability over a given period, which refers to the amount of real money (in percentage equivalent) that this investment has generated (Bodie et al., 2020; Elton et al., 2003). Both parameters of mathematics are actively used for this analysis. On the one hand, probability can provide valuable insights into which of their investment strategies are profitable (Bodie et al., 2020). For example, purchasing real estate for investment involves risk, as the calculation estimates the likelihood of selling the house after some time; this indicator can change depending on external conditions.
On the other hand, calculating average rates of return can be helpful when comparing different strategies to select the one with the highest return. Combinatorics can be helpful here, enabling the identification of various combinations of strategies that yield higher investment returns. Risk analysis is also closely related to the concepts under study since investments are primarily risks and, therefore, need to be calculated and determined.
Thus, mathematics applies to investment analysis because risk assessment, calculation of expected returns, and selection of more profitable strategies are based on the described calculations. Given the traditional multiplicity of financial data, probability and combinatorics can be utilized for computerized calculations, which create broader application advantages because, despite the increase in input data, the rules of calculation remain unchanged. The long-term perspective, however, is to increase personal returns, which is the goal of investing (Bodie et al., 2020). It is essential to realize that computational data does not provide an exact and unambiguous answer, as investment outcomes depend on various, sometimes unrelated variables. However, ignoring the math would drastically lower the chances of success and would be based solely on chance, rather than understanding the numbers and making informed choices.
Word Problem
The example of a verbal task can show the application of the described mathematical concepts. Specifically, it can be formulated as follows:
A financial advisor working with a client wants to create a diversified investment portfolio. The client has $200,000 and is considering three asset classes: stocks, bonds, and real estate investment trusts (REITs). Over the past ten years, returns have been positive in 4 years for stocks, six years for investments, and two years for REITs. The client wants to spread their investments across these asset classes while minimizing risk.
To accomplish this task, it is paramount to calculate the probabilities of return for each strategy. To do this, one can use the idea that the probability is the number of favorable outcomes (positive returns) divided by the total number of years. Then, the expected profitability for the three strategies is calculated as shown below:
p_stocks=4/10=0.4 [10]
p_obligations=6/10=0.6 [11]
p_REITs=2/10=0.2 [12]
Now that the calculations have been completed, it is essential to clarify that the results are averages, as the average annual profitability values for each of the ten years were involved. Comparing these average probabilities indicates, for example, that bonds may be a better investment option, giving a 60% rate of return. Connecting combinatorics to this problem is asset diversification (Esajian, 2022). The ordering of strategies is not essential, so permutation is not applicable. However, combinations are applicable: if one assumes that there are, for example, 15 stock options to choose from and only three need to be chosen, then according to Equation [8], the number of possible choices is:
Hence, the investor has 455 strategies to choose from, including stocks to invest in. Comparing their likely profitability can again help determine the best combination. However, risk assessment can help ignore stock choices that increase the riskiness of investing. To summarize, both branches of mathematics show that allocating the investment amount among different assets, when carefully analyzed, provides an advantage: reducing risk while increasing profitability.
Conclusion
Math is not limited to textbook pages and has the broadest real-life applications. Probability and counting are two important branches of mathematics that enable us to work with data, studying its distributions and forming valid conclusions. The applicability of these branches of math is clear for public health and investment management, as shown in the paper. In both cases, mathematics helps make more informed decisions and minimize societal or personal risks.
References
Andrews, N., Tessier, E., Stowe, J., Gower, C., Kirsebom, F., Simmons, R., & Bernal, J. L. (2021). Vaccine effectiveness and duration of protection of Comirnaty, Vaxzevria and Spikevax against mild and severe COVID-19 in the UK.
Bodie, Z., Kane, A., & Marcus, A. (2020). Investments. McGraw-Hill Education.
Brauer, F., Castillo-Chavez, C., & Feng, Z. (2019). Mathematical models in epidemiology (Vol. 32). Springer.
Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2003). Modern portfolio theory and investment analysis. Wiley.
Esajian, P. (2022). 6 types of investment strategies for beginners. Fortune Builders.