3.2. Axiomatic Definition of Probability

The Axiomatic Definition of probability was introduced by Andrey Kolmogorov in 1933. It provides a mathematically rigorous foundation for probability and can handle infinite sample spaces and non-uniform probabilities.

The Axiomatic Definition of probability, established by Kolmogorov, sets the groundwork for probability theory. Unlike earlier models, it is not limited to finite and equally likely outcomes. Instead, it can accommodate a wide array of scenarios, including those that involve an infinite number of possibilities and different likelihoods for those possibilities.

Imagine a vast library containing millions of books. Each book represents an outcome. In a classical definition, you might only consider the first 100 books (finite sample space). However, the axiomatic definition allows you to discuss probabilities concerning all books in the library, regardless of whether the collection is finite or infinite.

A probability space consists of three elements:

• Sample space (S): The set of all possible outcomes.
• Set of events (F): A collection of subsets of S, including S and the empty set.
• Probability function (P): A function that assigns a probability to each event in F.

A probability space is made up of three critical components. The sample space (S) encompasses all the outcomes of an experiment. The set of events (F) includes various groups of outcomes, which could be as simple as an event of rolling a three or as complex as defining multiple combinations of outcomes. Lastly, the probability function (P) quantitatively assigns probabilities to these events, facilitating calculations based on defined criteria.

Consider hosting a game night. Your sample space (S) consists of the various games you can play (like Monopoly, Chess, etc.). Events (F) refer to different game scenarios (playing Monopoly, or choosing card games). The probability function (P) is akin to how likely each game is to be chosen, depending on player preferences.

Let 𝑃:𝐹 → [0,1] be a probability function. Then it must satisfy: Axiom 1 (Non-negativity): 𝑃(𝐸) ≥ 0 for every event 𝐸 ∈ 𝐹 Axiom 2 (Normalization): 𝑃(𝑆) = 1 Axiom 3 (Additivity): If 𝐸1, 𝐸2, 𝐸3,… are mutually exclusive events, then: ∞𝑃(⋃𝐸𝑖) = ∑𝑃(𝐸𝑖) 𝑖=1.

Kolmogorov established three primary axioms that all probability functions should adhere to. The first axiom ensures that probabilities cannot be negative, as negative likelihoods don't make sense (Axiom 1). The second axiom asserts that when considering all possible outcomes, the total probability must equal one (Axiom 2), ensuring certainty. Lastly, the third axiom deals with the concept of mutually exclusive events, stating that the probability of any one of several mutually exclusive events occurring is the sum of their individual probabilities (Axiom 3).

Think of a school where students can only choose a single lunch option each day (mutually exclusive). Axiom 1 guarantees that the chance of a student going hungry (not eating) can't be less than 0. Axiom 2 means that if you total all lunch options' probabilities, they will equal the certainty that every student will eat something (1). Axiom 3 allows you to calculate the likelihood of a student choosing between pizza or pasta by adding their individual probabilities together.

Example: Tossing a fair coin Let S = {H, T} Define F = {∅, {H}, {T}, {H, T}} Assign P({H}) = 0.5, P({T}) = 0.5 This satisfies all three axioms:

• Non-negativity: 0.5 ≥ 0
• Normalization: P({H, T}) = 0.5 + 0.5 = 1
• Additivity: P({H} ∪ {T}) = P({H}) + P({T})

When tossing a fair coin, the sample space S consists of two outcomes: heads (H) and tails (T). The event space F includes every possible outcome, including nothing at all (empty set ∅) and the total outcomes (H and T together). The assigned probabilities satisfy Kolmogorov’s axioms: they are non-negative (both half), they sum up to one (certainty that something will happen), and they add correctly when considering the union of the two events.

Consider a simple decision-making process, such as flipping a coin to determine who goes first in a game. The outcomes (H and T) directly reflect the two possible results. The axioms ensure that you understand there’s a fair chance for either player to start the game, reinforcing fair play and equality in decision-making.

• Applicable to both finite and infinite sample spaces. • Handles both discrete and continuous cases. • Can model real-world probabilities where outcomes are not equally likely.

One of the primary strengths of the Axiomatic Definition of probability is its versatility. It can apply to finite sample spaces (like flipping a coin) and infinite spaces (like calculating probabilities based on natural numbers). It also effectively tackles both discrete situations (where outcomes can be counted) and continuous cases (like measuring time or distance). Importantly, it can model complex and realistic scenarios where outcomes do not possess equal likelihood, enhancing its usability across various fields.

Think of weather forecasting. The outcomes for a sunny, rainy, or snowy day don't have equal probabilities based on historical data. The axiomatic approach accommodates this imbalance, confidently providing the likelihood of a sunny day versus a snowy one based on empirical data, regardless of whether there are only a few outcomes or infinitely many combinations of weather factors at play.

The classical definition is a special case of the axiomatic definition where all outcomes are equally likely, and the sample space is finite.

The Axiomatic Definition encompasses the principles of the Classical Definition as a specific scenario. In cases where we deal with finite sample spaces and all outcomes have equal probabilities (like tossing a fair die), the classical approach naturally fits under the broader axiomatic framework. However, the axiomatic definition provides greater flexibility and applicability to more complex situations.

Consider flipping a coin again. While the classical definition applies perfectly here (0.5 for heads and 0.5 for tails), if we analyze the chance of drawing a card from a deck where there are varying chances for winning based on the suit, we step into the realm of axiomatic probability. This showcases how the broader framework not only includes the simpler classical perspective but expands to accommodate and explain the complexities of real-world probabilities.

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