1.3 Axioms of probability

Probability axioms are the building blocks of probability theory. They define how we measure and calculate chances of events happening. These rules help us make sense of uncertainty in everyday life and complex scientific scenarios.

Understanding these axioms is crucial for grasping more advanced probability concepts. They allow us to model real-world situations, from coin flips to stock market predictions, providing a solid foundation for statistical analysis and decision-making.

Axioms of Probability

Fundamental Axioms

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• First axiom establishes non-negativity of probability measures for any event A in sample space $P(A) \geq 0$
• Second axiom defines probability of entire sample space S equals 1 $P(S) = 1$
• Third axiom (countable additivity) states probability of union of mutually exclusive events equals sum of individual probabilities $P(A_1 \cup A_2 \cup ...) = P(A_1) + P(A_2) + ...$
• Axioms formalized by Andrey Kolmogorov in 1933 provide foundation for probability theory
• Apply to both discrete and continuous probability distributions creating unified framework

Implications and Importance

• Ensure consistency and mathematical well-definition of probability measures
• Crucial for deriving complex probability rules and theorems (law of large numbers, central limit theorem)
• Enable calculation of probabilities for compound events (unions, intersections)
• Facilitate development of statistical inference and hypothesis testing
• Allow for modeling of real-world phenomena with random components (quantum mechanics, financial markets)

Probability Calculations

Basic Probability Rules

• Complement rule calculates probability of event not occurring $P(A^c) = 1 - P(A)$ (finding probability of not rolling a 6 on a die)
• Addition rule for mutually exclusive events $P(A \cup B) = P(A) + P(B)$ when A and B are disjoint (probability of rolling an even number or a 5 on a die)
• General addition rule for non-mutually exclusive events $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (probability of drawing a heart or a face card from a standard deck)
• Multiplication rule for independent events $P(A \cap B) = P(A) \times P(B)$ (probability of getting heads on two consecutive coin flips)

Advanced Probability Calculations

• Conditional probability definition $P(A|B) = \frac{P(A \cap B)}{P(B)}$ where $P(B) > 0$ (probability of having a disease given a positive test result)
• Law of total probability computes probabilities involving partitions of sample space $P(A) = \sum_{i} P(A|B_i) P(B_i)$ (calculating overall probability of an event considering multiple scenarios)
• Bayes' theorem updates probabilities based on new information $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$ (updating probability of having a disease after receiving test results)
• Inclusion-exclusion principle generalizes addition rule for multiple events $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$ (calculating probability of at least one event occurring among multiple events)

Properties of Probability Measures

Fundamental Properties

• Probability measures bounded between 0 and 1 inclusive $0 \leq P(A) \leq 1$ for all events A
• Probability of empty set (impossible event) equals zero $P(\emptyset) = 0$
• Monotonicity property states if $A \subseteq B$, then $P(A) \leq P(B)$ (probability of drawing a king from a deck is less than or equal to probability of drawing a face card)
• Continuity from below for increasing sequence of events, limit of probabilities equals probability of limit $\lim_{n \to \infty} P(A_n) = P(\lim_{n \to \infty} A_n)$ (modeling convergence of probabilities in infinite sequences)

Advanced Properties and Applications

• Continuity from above for decreasing sequence of events, limit of probabilities equals probability of limit $\lim_{n \to \infty} P(A_n) = P(\lim_{n \to \infty} A_n)$ (analyzing behavior of probabilities in nested sequences of events)
• Properties enable derivation of advanced probabilistic concepts (law of large numbers, central limit theorem)
• Facilitate development of measure theory and its applications in probability and statistics
• Allow for rigorous treatment of continuous probability distributions and their properties
• Support construction of stochastic processes and their analysis (Markov chains, Brownian motion)