Glossary

2.1 Axioms of probability

3 min read•july 19, 2024

Probability axioms form the foundation of probability theory. These three fundamental rules—non-negativity, unitarity, and additivity—define how probabilities behave and allow us to calculate chances for various events.

Understanding these axioms is crucial for solving probability problems. They help us verify our calculations, combine probabilities for different events, and derive more complex rules. This knowledge is essential for tackling real-world scenarios in engineering, science, and finance.

Axioms of Probability

Axioms of probability

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  • Axiom 1: Non-negativity
    • States the probability of any A is greater than or equal to zero (P(A)≥0P(A) \geq 0)
    • Ensures probabilities are not negative (rolling a die, selecting a defective item)
  • Axiom 2: Unitarity
    • Asserts the probability of the entire S is equal to one (P(S)=1P(S) = 1)
    • Represents the idea that something must occur in a random experiment (flipping a coin, drawing a card)
  • Axiom 3: Additivity
    • For any two mutually exclusive events A and B, the probability of their union is equal to the sum of their individual probabilities (P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B), if A∩B=∅A \cap B = \emptyset)
    • Allows calculation of probabilities for combined events that cannot occur simultaneously (rolling an even or odd number on a die)

Application of probability axioms

  • Use Axiom 1 to verify all calculated probabilities are non-negative
    • Ensures results make sense in the context of the problem (probabilities of defective items, success rates)
  • Apply Axiom 2 to confirm the sum of probabilities for all outcomes in a sample space equals one
    • Helps validate probability distributions (rolling a die, spinning a wheel)
  • Utilize Axiom 3 to calculate the probability of the union of mutually exclusive events by summing their individual probabilities
    • Simplifies computation for events that cannot occur together (drawing a red or black card, selecting a male or female student)
  • Extend Axiom 3 to more than two mutually exclusive events: P(A1∪A2∪...∪An)=P(A1)+P(A2)+...+P(An)P(A_1 \cup A_2 \cup ... \cup A_n) = P(A_1) + P(A_2) + ... + P(A_n), if Ai∩Aj=∅A_i \cap A_j = \emptyset for all i≠ji \neq j
    • Generalizes the additivity property for multiple events (rolling a 1, 2, or 3 on a die)

Properties from probability axioms

  • Complement rule: The probability of an event A and its complement A' sum to one (P(A)+P(A′)=1P(A) + P(A') = 1)
    • Useful for calculating probabilities of complementary events (probability of not rolling a 6 on a die)
  • Monotonicity: If event A is a subset of event B, then the probability of A is less than or equal to the probability of B (P(A)≤P(B)P(A) \leq P(B), if A⊆BA \subseteq B)
    • Formalizes the idea that a more specific event cannot be more likely than a broader event containing it (probability of drawing a heart is less than or equal to drawing a red card)
  • Inclusion-exclusion principle for two events: P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
    • Calculates the probability of the union of two events by accounting for their overlap (probability of selecting a student who plays sports or music)
  • Generalized inclusion-exclusion principle for n events: P(⋃i=1nAi)=∑i=1nP(Ai)−∑i<jP(Ai∩Aj)+∑i<j<kP(Ai∩Aj∩Ak)−...+(−1)n+1P(A1∩A2∩...∩An)P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i) - \sum_{i<j} P(A_i \cap A_j) + \sum_{i<j<k} P(A_i \cap A_j \cap A_k) - ... + (-1)^{n+1} P(A_1 \cap A_2 \cap ... \cap A_n)
    • Extends the inclusion-exclusion principle to multiple events (probability of a product having at least one of several defects)

Scenarios for probability axioms

  • Axioms of probability apply to any well-defined random experiment with a clear sample space and event space
    • Rolling dice: Sample space is {1, 2, 3, 4, 5, 6}, events can be defined as subsets (rolling an even number)
    • Flipping coins: Sample space is {H, T}, events can be combinations of outcomes (flipping at least one head in three tosses)
    • Drawing cards from a deck: Sample space is the 52 cards, events can be specific cards or suits (drawing a red face card)
    • Selecting items from a manufacturing process: Sample space is all produced items, events can be defined based on quality criteria (selecting a defective item)
  • Axioms serve as the foundation for deriving other probability rules and properties
    • Enable the development of more complex probability concepts (conditional probability, independence)
  • Axioms are essential for solving a wide range of probability problems in various fields
    • Engineering: Reliability analysis, quality control
    • Science: Quantum mechanics, genetics
    • Finance: Portfolio optimization,

Key Terms to Review (20)

Additivity Axiom: The additivity axiom is a fundamental principle in probability theory stating that for any two mutually exclusive events, the probability of their union is equal to the sum of their individual probabilities. This principle highlights the relationship between different events and provides a foundational rule for calculating probabilities in various scenarios involving outcomes that cannot occur simultaneously.
Bayes' Theorem: Bayes' Theorem is a mathematical formula that describes how to update the probability of a hypothesis based on new evidence. It connects prior knowledge with new data to provide a revised probability, making it essential in understanding conditional probabilities and decision-making processes under uncertainty.
Binomial Distribution: The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. This distribution is essential for understanding random events that have two possible outcomes, like flipping a coin or passing a test, and it connects closely with the foundational concepts of probability, randomness, and statistical measures.
Combination: A combination is a selection of items from a larger set, where the order of selection does not matter. This concept is essential in understanding how to count outcomes in probability, particularly when determining the likelihood of certain events occurring. It contrasts with permutations, where the arrangement of items is significant, and helps in applying the axioms of probability to derive meaningful results from discrete sample spaces.
Complement of an Event: The complement of an event consists of all the outcomes in a sample space that are not included in the event itself. Understanding this concept is crucial for calculating probabilities since the complement helps in determining the likelihood of an event not occurring, reinforcing the relationship between events, sample spaces, and overall probability calculations.
Continuous Probability: Continuous probability refers to the likelihood of outcomes that can take any value within a specified range, as opposed to discrete outcomes which are distinct and separate. This concept is crucial in understanding how probabilities are assigned to intervals of values rather than specific points. It relies on the use of probability density functions (PDFs) to represent probabilities over continuous intervals, which connects to fundamental principles and methodologies in probability theory.
Counting Principle: The counting principle is a fundamental concept in combinatorics that provides a systematic method for counting the number of ways an event can occur. It states that if one event can occur in 'm' ways and a second independent event can occur in 'n' ways, then the total number of ways both events can occur is the product of the two, which is 'm * n'. This principle is essential for understanding the basic concepts of probability and randomness, as well as forming the basis for calculating probabilities based on different events.
Dependent Events: Dependent events are situations where the outcome or occurrence of one event affects the outcome or occurrence of another event. This relationship is crucial in understanding probability, as it highlights how events can influence each other rather than being completely independent. Recognizing dependent events is essential for calculating joint probabilities and applying the axioms of probability effectively.
Discrete Probability: Discrete probability refers to the probability of outcomes in a discrete sample space, where outcomes are countable and distinct. This type of probability deals with events that can take on specific values, such as the roll of a die or the number of heads in a series of coin flips. Understanding discrete probability is essential for applying the axioms of probability, calculating conditional probabilities, and utilizing Bayes' theorem effectively.
Event: In probability, an event is a specific outcome or a set of outcomes from a random experiment. Events are essential because they help define what we are interested in measuring, analyzing, or predicting in a random process. Understanding events allows us to connect various aspects like sample spaces, which list all possible outcomes, and probability models that describe how likely events are to occur.
Failure Analysis: Failure analysis is the process of investigating and understanding the reasons behind a system or component's failure in order to prevent future occurrences. This concept is crucial in various fields, especially engineering and manufacturing, as it helps identify patterns of failure that can be linked to probabilistic events and distributions. By employing statistical methods, this analysis connects the dots between failure events and underlying probabilities, enhancing reliability through informed decision-making.
Independent Events: Independent events are occurrences in probability where the outcome of one event does not affect the outcome of another. This concept is fundamental in understanding probability and randomness, as it allows for the simplification of calculations and predictions when events are unrelated.
Law of Total Probability: The law of total probability states that the total probability of an event can be found by considering all possible ways that the event can occur, weighted by the probabilities of those ways. This concept connects to other important features such as the axioms that govern how probabilities are assigned and manipulated, independence which allows for simplifications in calculations, joint probability distributions that consider multiple random variables, and the relationship between marginal and conditional distributions which provides clarity on how probabilities interact.
Non-negativity Axiom: The non-negativity axiom is a fundamental principle in probability theory that states the probability of any event must be greater than or equal to zero. This means that no event can have a negative likelihood of occurring, reinforcing the idea that probabilities reflect the degree of belief in an event's occurrence within a defined sample space.
Normal Distribution: Normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions. This distribution is vital in various fields due to its properties, such as being defined entirely by its mean and standard deviation, and it forms the basis for statistical methods including hypothesis testing and confidence intervals.
Permutation: A permutation is an arrangement of objects in a specific order, and it is essential in counting and probability scenarios. Understanding permutations allows for the evaluation of different possible outcomes when order matters, making it critical for calculations related to events and their probabilities. This concept is foundational when exploring the axioms of probability, as it helps to determine how many ways outcomes can occur.
Risk Assessment: Risk assessment is the systematic process of identifying, evaluating, and prioritizing risks associated with uncertain events or conditions. This process is essential in understanding potential negative outcomes, which can inform decision-making and resource allocation in various contexts such as engineering, finance, and healthcare.
Sample Space: A sample space is the set of all possible outcomes of a random experiment. It serves as the foundation for probability, helping us understand what outcomes we might encounter and how to analyze them. By identifying the sample space, we can define events and outcomes more clearly, which is essential when constructing probability models and interpretations, and helps in applying the axioms of probability along with set theory and operations.
Uniform Distribution: Uniform distribution is a type of probability distribution in which all outcomes are equally likely within a specified range. This means that every interval of the same length within the range has the same probability of occurring, making it a fundamental concept in understanding randomness and variability.
Unit Measure Axiom: The unit measure axiom states that the probability of the entire sample space is equal to one. This means that when considering all possible outcomes of a random experiment, the total probability must sum up to one, ensuring a complete representation of all possibilities. This axiom is fundamental in establishing a coherent probability model, as it lays the groundwork for calculating probabilities of specific events within the sample space.
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Glossary
Glossary
2.2 Set theory and operations