Fundamental Axioms of Probability
Basic probability is governed by three axioms proposed by Andrey Kolmogorov. First, the probability of any event is a number between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. Second, the probability of the entire sample space is 1. Third, for mutually exclusive events, the probability of their union is the sum of their individual probabilities. These axioms form the foundation for all probability calculations.
Addition Rule for Probabilities
The addition rule states that the probability of the union of two events A or B is P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually exclusive events, where P(A ∩ B) = 0, it simplifies to P(A ∪ B) = P(A) + P(B). This rule is essential for calculating probabilities of combined outcomes, such as the chance of drawing a red or black card from a deck.
Multiplication Rule and Conditional Probability
The multiplication rule for the intersection of events is P(A ∩ B) = P(A) * P(B|A), where P(B|A) is the conditional probability of B given A. If events are independent, P(B|A) = P(B), simplifying to P(A ∩ B) = P(A) * P(B). For example, in rolling two dice, the probability of getting a 1 on both (independent events) is (1/6) * (1/6) = 1/36.
Applications and Importance
These rules underpin decision-making in fields like statistics, economics, and science. They enable predictions in risk assessment, such as insurance premiums based on accident probabilities, or in genetics for inheritance patterns. Understanding them helps avoid errors in interpreting data and supports informed choices in uncertain scenarios.