Theoretical probability

What Is Theoretical Probability?

Theoretical probability is a concept within quantitative analysis that calculates the likelihood of an event occurring based on reasoning about the possible outcomes, rather than on observed data from experiments. It assumes that all potential outcomes are equally likely. This branch of statistics provides a mathematical framework for understanding uncertainty and is a foundational element in fields ranging from games of chance to complex financial modeling. Theoretical probability helps to establish an expected value for an event, serving as a benchmark for comparison with actual results.

History and Origin

The formal study of probability, including theoretical probability, began in the 17th century, largely spurred by a correspondence between French mathematicians Pierre de Fermat and Blaise Pascal in 1654. Their work was inspired by problems related to games of chance, such as how to fairly divide stakes in an interrupted game.²² Prior to this, rudimentary concepts of chance existed, but Fermat and Pascal laid the groundwork for modern probability theory, defining symmetrical cases and the comparison of favorable outcomes to total possibilities.,²¹ This initial work, sometimes referred to as "the doctrine of chances," expanded beyond gambling to areas like demographics and insurance, with contributions from Jakob Bernoulli and Pierre-Simon Laplace.²⁰

Key Takeaways

• Theoretical probability determines the likelihood of an event based on logical reasoning and the assumption of equally likely outcomes.
• It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
• Unlike empirical probability, theoretical probability does not rely on actual experiments or observed data.
• Its applications span various fields, including finance, risk management, and scientific research.
• Theoretical probability provides a baseline expectation against which real-world results can be compared.

Formula and Calculation

The formula for theoretical probability (P) of an event (E) is expressed as:

$P(E) = \frac{\text{Number of favorable outcomes for E}}{\text{Total number of possible outcomes}}$

Where:

• P(E) represents the probability of event E occurring.
• Number of favorable outcomes for E refers to the count of ways event E can happen.
• Total number of possible outcomes refers to the total count of all equally likely results in the sample space.

For example, when rolling a fair six-sided die, the total number of possible outcomes is 6 (sides 1, 2, 3, 4, 5, 6). If the event E is rolling an even number, the favorable outcomes are 2, 4, and 6, so there are 3 favorable outcomes.
Thus, the theoretical probability of rolling an even number is:

$P(\text{Even number}) = \frac{3}{6} = 0.5$

This calculation provides a fixed likelihood based purely on the properties of the die and the nature of the event.

Interpreting Theoretical Probability

Theoretical probability provides a quantifiable measure, typically a value between 0 and 1 (or 0% and 100%), representing the inherent likelihood of an event. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event. For instance, a coin flip has a theoretical probability of 0.5 for heads, assuming a fair coin. This interpretation is crucial in many areas, including decision making under uncertainty, where understanding the intrinsic likelihood of various scenarios is paramount. It forms the basis for predicting the long-term frequency of events if an experiment were to be repeated an infinite number of times under ideal conditions.

Hypothetical Example

Consider a hypothetical financial analyst assessing the probability of a specific stock price movement. While real-world stock movements are complex, a simplified model can illustrate theoretical probability. Suppose an analyst wants to determine the probability of a company's stock price, currently at $100, either staying the same or increasing to $101 over a very short period, given that it can only move to $99, $100, or $101, and each outcome is considered equally likely for this simplified scenario.

1. Define the event: The event (E) is the stock price either staying the same or increasing.
2. Identify favorable outcomes: The favorable outcomes are the stock price remaining at $100 or increasing to $101. (2 outcomes)
3. Identify total possible outcomes: The total possible outcomes are $99, $100, and $101. (3 outcomes)
4. Calculate theoretical probability: Using the formula, the theoretical probability is the number of favorable outcomes divided by the total number of possible outcomes.
   $P(E) = \frac{2}{3} \approx 0.67$
   This simplified example, while not reflecting the complexities of actual financial markets, demonstrates how theoretical probability can be applied to define expected outcomes within a structured, limited set of possibilities, providing a foundational understanding before incorporating more nuanced risk factors.

Practical Applications

Theoretical probability serves as a bedrock for various practical applications across finance and other disciplines, particularly in areas involving risk management and quantitative analysis.

• Financial Modeling: In financial modeling, theoretical probability underpins many valuation models, such as the Black-Scholes model for options pricing, which uses a probability distribution to estimate the likelihood of various stock prices at option expiration.¹⁹
• Actuarial Science: Actuaries use theoretical probability to calculate premiums for insurance policies, assessing the likelihood of events like accidents, illnesses, or deaths to ensure solvency and fair pricing.¹⁸
• Investment Portfolio Optimization: Modern portfolio theory utilizes theoretical probability to evaluate the expected returns and risks of different assets, helping investors construct diversified portfolios that align with their risk tolerance.¹⁷
• Algorithmic Trading: Algorithmic trading strategies often incorporate theoretical probability to model market behavior and identify high-probability trading opportunities.
• Regulatory Frameworks: Regulatory bodies like the Federal Reserve use quantitative risk analysis, which heavily relies on probability theory, to assess the capital adequacy and financial stability of institutions, particularly through stress testing.¹⁶,¹⁵ This helps in understanding potential impacts of adverse market movements, though models are simplifications of reality.¹⁴

These applications demonstrate how theoretical probability moves beyond abstract concepts to inform concrete financial strategies and regulatory oversight. The Massachusetts Institute of Technology (MIT) Technology Review has explored how mathematical approaches help in predicting stock market futures, highlighting the real-world utility of these probabilistic frameworks.¹³

Limitations and Criticisms

While theoretical probability is a powerful tool, it operates under significant assumptions that limit its applicability, especially in complex systems like financial markets. A primary criticism is its reliance on the assumption of equally likely outcomes, which rarely holds true in dynamic, real-world scenarios. For instance, a coin might not be perfectly fair, or a die might be weighted, making actual outcomes deviate from theoretical expectations.¹²

Furthermore, theoretical probability models struggle with epistemic uncertainty—situations where the set of all possible outcomes is unknown or evolves over time. T¹¹his contrasts with aleatoric uncertainty, which refers to inherent randomness within a fixed set of known outcomes. F¹⁰inancial markets, for example, are influenced by countless unpredictable factors, including human behavior, geopolitical events, and technological advancements, which theoretical models may not fully capture. This limitation means that unexpected "black swan" events, which fall outside conventional probability distributions, are often underestimated or ignored.

⁹The Federal Reserve Bank of San Francisco has discussed that while quantitative risk management models provide a coherent framework for identifying and analyzing risks, they are simplifications of reality and may not capture every aspect, particularly unlikely yet possible events that could cause significant losses. T⁸herefore, while theoretical probability provides a foundational understanding, its direct application requires careful consideration of its underlying assumptions and the inherent volatility and complexity of the financial landscape.

⁷## Theoretical Probability vs. Empirical Probability

Theoretical probability and empirical probability are two fundamental approaches to quantifying the likelihood of events, often confused despite their distinct methodologies. The core difference lies in their basis: theoretical probability is derived from logical reasoning about equally likely outcomes, while empirical probability is based on actual observations from experiments or historical data.,,^6
5
4| Feature | Theoretical Probability | Empirical Probability |
| :-------------------- | :----------------------------------------------------------- | :----------------------------------------------------------- |
| Basis | Logic, inherent properties of the event | Observed data, results of experiments or past events |
| Calculation | (Favorable Outcomes) / (Total Possible Outcomes) | (Number of Times Event Occurred) / (Total Number of Trials) |
| Assumptions | All outcomes are equally likely, ideal conditions | Reflects real-world conditions, may not be ideal or perfectly random |
| Prediction | Expected long-term frequency under ideal conditions | Observed frequency based on actual trials; can vary with trials |
| Relationship | As the number of trials increases, empirical probability tends to converge towards theoretical probability, demonstrating the Law of Large Numbers. |³
| Use Cases | Games of chance, conceptual modeling, initial risk assessment | Real-world data analysis, historical performance, backtesting strategies |

While theoretical probability offers an ideal benchmark, empirical analysis provides insights into real-world behavior, which may deviate due to imperfect conditions or random variables., ²B¹oth forms of probability are crucial for a complete understanding of data analysis and risk assessment.

FAQs

What does "equally likely outcomes" mean in theoretical probability?

"Equally likely outcomes" means that each possible result of an event has the same chance of occurring. For example, when flipping a fair coin, the outcome of "heads" is just as likely as "tails." In theoretical probability, calculations assume this perfect fairness, even if real-world conditions might introduce slight biases.

Can theoretical probability be applied to real-world financial markets?

Directly applying theoretical probability to real-world financial markets can be challenging due to their complexity and the presence of factors not easily accounted for, such as behavioral economics and unforeseen events. However, it serves as a foundational concept in financial modeling and statistical inference, where idealized conditions are used to build models that are then refined with real data.

Is a theoretical probability of 0.5 always the same as a 50% chance?

Yes, a theoretical probability of 0.5 is equivalent to a 50% chance. Probability values are typically expressed as decimals between 0 and 1, where 1 represents 100% certainty, and 0 represents 0% certainty. Multiplying the decimal probability by 100 converts it to a percentage.

Why might theoretical probability not match actual results in a short series of events?

Theoretical probability describes what is expected over a very large number of trials. In a short series of events, actual results can deviate significantly due to pure chance. For example, flipping a fair coin four times theoretically should yield two heads and two tails, but it's common to see results like three heads and one tail. The Law of Large Numbers states that as the number of trials increases, the observed (empirical) probability will tend to converge towards the theoretical probability.

What's the significance of theoretical probability in assessing investment opportunities?

Theoretical probability helps investors understand the fundamental likelihood of certain outcomes if conditions were ideal. While investment outcomes are rarely ideal, it provides a baseline for comparing different opportunities and for constructing models that estimate potential returns and risks. It is a critical component in understanding concepts like expected return for an asset.