Non-negativity, additivity, and the property that the total probability sums to 1

Introduction

Probability theory provides a rigorous framework for quantifying uncertainty, and its foundational principles—non-negativity, additivity, and the normalization condition—ensure consistency and coherence in probabilistic reasoning. These properties are not merely technical constraints but are essential to the development of probability models, from simple experiments to complex stochastic processes. Non-negativity ensures that probabilities are always positive, additivity guarantees the compatibility of mutually exclusive events, and the total probability sum to 1 ensures that the entire sample space is accounted for. Together, these properties underpin the axioms of probability and are critical to the validity of any probabilistic analysis.

Non-negativity

In probability theory, non-negativity refers to the requirement that for any event $ A $, the probability $ P(A) $ satisfies $ 0 \leq P(A) \leq 1 $. This condition is fundamental to the definition of a probability measure, which assigns to each event a value between 0 and 1. The non-negativity property ensures that probabilities are not assigned to impossible events (which would have zero probability) or to events with negative likelihoods, which are logically inconsistent.

The mathematical expression of non-negativity is straightforward:
$$ 0 \leq P(A) \leq 1 \quad \text{for all events } A. $$
This property is particularly crucial in practical applications, such as calculating the likelihood of outcomes in dice rolls, coin flips, or random number generators. For example, when rolling a fair six-sided die, the probability of rolling a 1 is $ \frac{1}{6} $, and the probability of rolling any number from 1 to 6 is $ \frac{1}{6} $, ensuring that all outcomes are non-negative and sum to 1.

Non-negativity also plays a key role in measure theory, where probabilities are treated as measures on a sample space. This ensures that the measure is well-defined and avoids contradictions. For instance, in the context of Lebesgue integration, non-negativity guarantees that the integral of a function over a domain is a non-negative quantity, aligning with the intuitive understanding of probability as a measure of likelihood.

Additivity

Additivity is another cornerstone of probability theory, stating that the probability of the union of any finite number of mutually exclusive events is equal to the sum of their individual probabilities. Formally, for any events $ A_1, A_2, \dots, A_n $ that are mutually exclusive, the following holds:
$$ P\left(\bigcup_{i=1}^{n} A_i\right) = \sum_{i=1}^{n} P(A_i). $$
This property is critical in scenarios where events are disjoint, such as rolling a die and observing a specific face, or selecting a random element from a set. For example, if a die is rolled, the probability of rolling a 1 or a 2 is $ \frac{1}{6} + \frac{1}{6} = \frac{1}{3} $, demonstrating additivity.

Additivity also extends to more complex scenarios, such as the addition of probabilities in independent events. If two events $ A $ and $ B $ are independent, the probability of their union is given by:
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B). $$
This formula reflects the principle that the probability of overlapping events must be subtracted to avoid double-counting. The additivity property is further reinforced by the concept of measure theory, where probabilities are treated as measures on a sample space, ensuring that the measure is additive over disjoint events.

The Total Probability Sums to 1 Property

The final property of probability theory is the normalization condition, which states that the total probability of all possible outcomes in a sample space must equal