1. Introduction

In the contemporary economic landscape, the unpredictability of market forces has become a significant concern. The ability to quantify and comprehend the randomness inherent in economic phenomena is therefore of paramount importance. The application of probability theory in economics allows for the study and explanation of uncertainty and randomness in economic activities. This uncertainty arises from a multitude of factors, including market fluctuations, policy changes, and technological innovations. By modeling and analyzing these randomnesses, economic probability theory can assist in the understanding of the evolution of economic phenomena and the modeling and analysis of them for entrepreneurs and investors. The paper aims to examine the scenarios of probability in economics and to predict and analyze the volatility of financial markets using the methods of economic probability modeling. The understanding of market volatility helps in predicting future market trends, which in turn helps investors and institutions manage risk and make decisions effectively. To explore the application of probability theory in economics, this paper employs a literature summary approach to study the basic computational methods of probability theory and some data modeling scenarios in the economic field. The analysis of these data can assist investors and entrepreneurs in making more informed decisions regarding future market trends, thereby enabling them to make the most appropriate choices.

2. Fundamentals of Probability

Probability statistics, sometimes referred to as mathematical statistics, is a mathematical approach to researching the statistical principles governing random events in nature. One basis is probability. The most efficient hand for carrying out quantitative research is the standard teaching instrument. To begin with, a deterministic phenomenon requires that a set of circumstances result in a proven outcome. For instance, water heated to room temperature and standard air pressure of 100 °C will undoubtedly boil. The second is uncertainty, which denotes that an image’s result is subject to change in specific circumstances. And if a coin is tossed, it may be heads or tails. This manifests itself in a large number of homogeneous random phenomena, and the resulting collective regularity is known as probability statistics [1].

2.1. Basic Concepts and Properties of Probability

Probability has values between 0 and 1, where 0 denotes impossibility and 1 denotes certainty. It is a quantitative measure of the likelihood of an event occurring. The collection of every conceivable result in a random experiment is referred to as the sample space. For example, the sample space for rolling a die is {1, 2, 3, 4, 5, 6}. During an experiment, an event might represent a single outcome or a group of outcomes inside the sample space. Take, for instance, the scenario in which a die roll yields an even number; this can be shown as {2, 4, 6}. P(A) is the probability that an event will occur, where A is the particular occurrence under consideration. In terms of probability, the range is 0 to 1, and its non-negative value is P(A) ≥ 0. Values close to 1 indicate certainty, and values close to 0 indicate improbability.

P(A and B)=P(AB)=P(A)P(B) is the total probability of two independent events, A and B [2]. When two events, A or B, may only happen one at a time, they are said to be mutually exclusive. When two occurrences are mutually exclusive, the probability that they will occur is denoted by P(AB) and P(A and B)=P(AB)=0. The probabilities of two events occurring independently are denoted by P(AB) and P(A or B), respectively, when they are mutually exclusive. AB = P(A) + P(B) - P(AB) = P(A) + P(B) - 0 = P(A) + P(B) [3]. For a scenario where the occurrences are not mutually exclusive, P(A or B) = P(AB) = P(A) + P(B) - P(A and B) or P(AB) = P(A) + P(B) - P(AB) [4].

The Bayes theorem is a technique that uses conditional probability computations to update prior values into posterior probabilities. P(A|B) is the probability of event A in the event that event B has occurred, and P(B|A) is the probability of event B in the event that event A has occurred. This formula can be found in P(B) = [P(B|A) × P(A)]/P(B). The prior probabilities are denoted by P(A) and P(B), respectively. These fundamental concepts and tenets serve as the cornerstone of probability theory, offering essential tools and theoretical support for understanding and interpreting a broad range of chance-based phenomena.

2.2. Conceptual Algorithm

The probability theory is represented by the axioms listed below: The following is a summary of the presumptions that guided the creation of the axioms: Let \( (Ω,F,P) \) be a measurable space where P(Ω)=1 and P(E) represents the probability of an occurrence E. The sample space (Ω), the event space (F), and the probability measure P make up the triple that is defined as the probability space (Ω,F,P) in this context. [5]

The first axiom is that the probability of an event is a non-negative real number:

\( P(E)∈R, P(E)≥0 ∀E∈F \) (1)

where \( F \) is the event space. Unlike more broad measure theory, it follows (when paired with the second postulate) that P(E) is always finite. The first axiom is relaxed in theories that assign negative probability.

The second axiom states that the likelihood that at least one of the elementary events in the entire sample space will occur is 1, which is the unit measure assumption.

\( P(Ω) \) =1(2)

And the third axiom is the assumption of σ-additivity. There exists any countable sequence of disjoint sets (synonymous with mutually exclusive events) \( E1,E2,…. \) =1 that satisfies

\( P(\bigcup _{i=1}^{∞}{E_{i}})=\sum _{i=1}^{∞}P({E_{i}}) \) (3)

In some studies, only finite addable probability spaces are considered, where only a set algebra is needed instead of a σ-algebra [6]. In general, quasi-probability relaxes the third axiom.

3. Applications of Probability in Economy

Probability theory has a wide range of applications in economic forecasting, especially in the prediction of future market trends, risk management, and decision-making. This chapter will explore the application of probability theory in economics and elaborate on its importance.

3.1. Application of Probabilistic Models in Market Forecasting

Probabilistic models play a pivotal role in market forecasting. In the case of the Turkish economy, for instance, a cointegrating vector autoregressive (VAR) model can be employed for market forecasting. The model is constructed in a manner analogous to the cointegration VAR-X model employed for the UK and Swiss economies. In order to construct the model, it is first assumed that the following three equilibrium relationships of the Turkish economy remain constant over a longer period of time [7].

\( {P_{t}}-{{P_{t}}^{*}}-{e_{t}}={a_{10}}+{a_{11}}t+{ε_{1,t}} \) (4)

\( {r_{t}}-{{r_{t}}^{*}}={a_{20}}+{ε_{2,t}} \) (5)

\( {y_{t}}-{{y_{t}}^{*}}={a_{30}}+{a_{31}}t+{ε_{3,t}} \) (6)

where ε_i,t+1, i = 1,2,3 are stationary errors representing the departures from long-run equilibriums predicted by related relations, and t is the linear trend. The relationship between international goods-market arbitrage and purchasing power parity (PPP) is shown in equation (7). The existence of a linear trend in the PPP connection may be justified by the Harrod-Samuelson-Balassa effects and/or measurement error in pricing, particularly when it comes to the handling of quality, but its validity still has to be empirically confirmed. Similarly, the interest rate parity relationship is defined by Equation (8), which takes into account the arbitrage between holdings of domestic and foreign bonds. Equation (9) employs a stochastic adaptation of the Solow growth model to depict an “output gap” (OG) relationship. The empirical validity of this relationship—that is, the stationarity of the Turkish output gap—may lend credence to the long-run convergence hypothesis between the output levels of the OECD and Turkey. In this instance, as with the PPP relation, empirical proof is necessary to confirm the existence of a linear trend in the OG connection. Indeed, depending on varying values of a30 and a31, distinct hypotheses for the March–April 2014 217convergence can be developed. A30 and A31 represent tight and speedy catching up, respectively [8]. In contrast, a30 ≠ 0 and a31 = 0 can represent rigorous or quick convergence. Finally, a30 = 0 and a31 = 0 can be used to characterize the zero-mean convergence scenario of Bernard and Durlauf [9]. The three long-run relationships (7)-(8) of the model can be succinctly expressed as follows.

\( {ε_{t}}={β^{ \prime }}{z_{t-1}}-{a_{1}}(t-1)-{a_{0}} \) (7)

where the parameters can be understood as follows:

\( {z_{t}}=({p_{t}},{e_{t}},{r_{t}}{{r_{t}}^{*}},{y_{t}},{{y_{t}}^{*}},{{p_{t}}^{t}}) \prime \)

\( {a_{t}}=({a_{11}},0,{a_{30}}) \prime \)

\( {a_{t}}=({a_{11}},{a_{20}},{a_{30}}) \prime \)

\( {ε_{t}}=({ε_{1,t}},{ε_{2,t}},{ε_{3,t}}) \prime \) (8)

And \( β \prime \) is expressed as

\( β \prime =(\begin{matrix}1 & -1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{matrix} \begin{matrix}0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 1 & -1 \\ \end{matrix} \begin{matrix}-1 \\ 0 \\ 0 \\ \end{matrix}) \) (9)

All the overidentifying constraints required to align with the long-term relationships are enforced by the matrix β'.

In this setup, the variables are partitioned such that \( {y_{t}}=({p_{t}}{,e_{t}},{r_{t}},{y_{t}}) \) is treated as an I (1) vector of endogenous variables, while x, = ( \( {{y_{t}}^{*}},{{p_{t}}^{*}},{{r_{t}}^{*}} \) ) is considered as an I(1) vector of weakly exogenous variables, where changes in weakly exogenous variables directly affect y but are not affected by the imbalances in the Turkish economy, which are measured by the error correction term. This standard small open economy setup seems to be the more natural choice.

The conditional error correction model that follows can be used to estimate parameters if weakly exogenous variables are assumed.

\( ∆{y_{t}}={a_{y}}-{α_{y}}[{β^{ \prime {z_{t-1}}}}-{a_{1}}(t-1)]+\sum _{i=1}^{p-1}{ψ_{yi}}∆{z_{t-i}}+{ψ_{yx}}∆{x_{t}}+{v_{yt}} \) (10)

where \( {v_{yt}} \) is a 4 x 3 matrix of serially uncorrelated shocks, \( {ψ_{yi}} \) is 4 x 7 matrix of short-run coefficients, \( {ψ_{yx}} \) is a 4 x 1 vector of coefficients capturing the effects of changes in the exogenous variables on \( ∆{y_{t}} \) .

To generate endogenous variable forecasts using conditional models, it is necessary to forecast exogenous variables. The following marginal model is specified to create exogenous variable predictions:

\( ∆{x_{t}}={a_{x}}+\sum _{i=1}^{k-1}{ψ_{xi}}∆{x_{t-1}}+{v_{xt}} \) (11)

where \( {v_{yt}} \) is a 3 \( × \) 1 shock vector that is considered uncorrelated with \( {v_{yt}} \) ; \( {a_{x}} \) is a 3 \( × \) 1 intercept vector; and \( {ψ_{xi}} \) is a 1 \( × \) 3 fixed coefficient vector.

Equations (10) and (11) combined with a solution for ∆ \( {z_{t}} \) yield the following vector error correction model:

\( ∆{z_{t}}=a-α[{β^{ \prime {z_{t-1}}}}-{a_{1}}(t-1)]+\sum _{i=1}^{p-1}{Γ_{i}}∆{z_{t-i}}+{u_{t}} \) (12)

where \( a = ({a_{x}},a_{y}^{ \prime }-{a_{x}} \prime {ψ_{yx}} \prime ): a = (0, {a_{y}} \prime ) \) , \( {Γ_{i}} \) = \( (ψ_{xi}^{ \prime },ψ_{yi}^{ \prime }-{ψ_{xi}} \prime {ψ_{yx}} \prime ) \) , and \( {u_{t}} \) ( \( {v_{xt}},{v \prime _{yt}}-{v \prime _{yt}}{ψ \prime _{yx}} \) ) is supposed to be IID (0, 2), the simplified form error vector, where 2 means the positive definite covariance matrix [7].

3.2. Follow-up Research Directions

Though significant progress has been made in the application of probabilistic models in the economy, there are still many future research directions to be explored. For example, the comparative advantages and disadvantages of different types of probabilistic models in market forecasting can be further investigated, as well as how probability theory can be better utilized to solve problems in the real economy. In addition, combining probability models with other methods can be considered to improve the accuracy and efficiency of market forecasting. By conducting further research on the application of probability models in the economy, the laws behind economic phenomena can be better understood.

4. Future Outlook

The future application of probability theory in economics holds considerable promise, driven by advancements in data analytics, computational power, and the increasing complexity of global financial markets. The integration of probability theory with advanced data analytics and machine learning is expected to enhance market prediction capabilities significantly. These technologies can process vast amounts of data in real time, identifying patterns and trends that traditional methods may overlook, leading to more precise predictions of market fluctuations and informed investment strategies.

Improved risk management is another key area where probability theory will play a crucial role. The deeper integration of probability theory with real-time data analytics will enable financial institutions to better assess and mitigate risks associated with market volatility, economic downturns, and global events. Probabilistic models will help develop robust strategies for capital allocation, stress testing, and contingency planning, ensuring a more resilient financial system. In the realm of personalized financial services, probability theory will enable institutions to offer tailored investment advice, credit scoring, and insurance products by analyzing individual behavioral data and market trends. This level of personalization will enhance customer satisfaction and optimize the allocation of financial resources.

Policymakers will increasingly rely on probability-based models to inform economic policy and planning. These models can simulate the impact of various policy decisions under different economic scenarios, providing a comprehensive understanding of potential outcomes and aiding in the formulation of resilient policies adaptable to changing economic conditions. Furthermore, by embracing probabilistic models and integrating them with emerging technologies, economists and financial professionals can navigate modern economic uncertainties more effectively. This approach will lead to better-informed decisions, enhanced risk management, and more resilient economic systems.

5. Conclusion

The summarizes the application of probability theory in economics, focusing on the importance of probability theory in risk management, investment decision-making and market forecasting, and points out the importance and impact of probability on decision-making. Probability theory provides theoretical support and practical guidance for decision making and helps to improve the efficiency of resource allocation and policy making. Therefore, the research significance of this paper is to delve into the impact of probability theory on decision making in a market economy and to highlight the importance of probability theory in shaping economic outcomes.

However, this experiment has some limitations. First, the construction of the paper relies heavily on the existing literature, and actual experiments are insufficient. Moreover, other possibilities are not taken into account, which may lead to bias in practical applications. In order to increase the reliability of the results, it would be beneficial to include additional factors, such as force majeure, in the analysis. In addition, conducting personal experiments could demonstrate more concretely the usefulness of probability in these areas. Future applications of probability are not limited to the economic field. Furthermore, probability can be applied to other fields such as climate observation and industrial production. In addition, research on different applications of probability and different probabilistic algorithms can be carried out to expand the scope of probability.