2.3 Properties of probability

Probability axioms are the foundation of probability theory. They define the rules for calculating probabilities and ensure consistency in our calculations. These axioms lead to important properties that help us solve complex probability problems.

Mutually exclusive and independent events are key concepts in probability. Understanding the difference between them is crucial for correctly applying probability rules and avoiding common mistakes in calculations.

Probability Axioms and Properties

Properties of probability axioms

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• Axiom 1: Non-negativity ensures probability of any event $A$ is greater than or equal to 0 ($P(A) \geq 0$)
• Axiom 2: Normalization sets probability of the entire sample space $S$ equal to 1 ($P(S) = 1$)
• Axiom 3: Countable additivity states for any sequence of mutually exclusive events $A_1, A_2, \ldots$, probability of their union equals the sum of their individual probabilities ($P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$)
• Properties derived from axioms include:
  • Probability of an impossible event (empty set $\emptyset$) equals 0 ($P(\emptyset) = 0$)
  • Probability of any event $A$ lies between 0 and 1 ($0 \leq P(A) \leq 1$)
  • Probability of the union of two events $A$ and $B$ equals the sum of their individual probabilities minus the probability of their intersection ($P(A \cup B) = P(A) + P(B) - P(A \cap B)$)

Addition and multiplication rules

• Addition rule calculates probability of the union of two events $A$ and $B$ ($P(A \cup B) = P(A) + P(B) - P(A \cap B)$)
  • For mutually exclusive events, probability of their union simplifies to the sum of their individual probabilities ($P(A \cup B) = P(A) + P(B)$)
• Multiplication rule calculates probability of the intersection of two events $A$ and $B$ ($P(A \cap B) = P(A) \cdot P(B|A)$)
  • For independent events, probability of their intersection equals the product of their individual probabilities ($P(A \cap B) = P(A) \cdot P(B)$)

Probability of event complements

• Complement of an event $A$, denoted as $A^c$ or $\bar{A}$, represents all outcomes in the sample space not included in $A$
• Probability of the complement of event $A$ equals 1 minus the probability of $A$ ($P(A^c) = 1 - P(A)$)

Mutually Exclusive and Independent Events

Mutually exclusive vs independent events

• Mutually exclusive events cannot occur simultaneously ($P(A \cap B) = 0$)
  • Probability of the union of mutually exclusive events $A$ and $B$ equals the sum of their individual probabilities ($P(A \cup B) = P(A) + P(B)$)
  • Examples of mutually exclusive events: rolling an even or odd number on a die, drawing a heart or a spade from a deck of cards
• Independent events occur without affecting each other's probability ($P(A \cap B) = P(A) \cdot P(B)$)
  • Probability of event $A$ given event $B$ equals the probability of $A$ ($P(A|B) = P(A)$) and vice versa ($P(B|A) = P(B)$)
  • Examples of independent events: flipping a coin and rolling a die, drawing a card and spinning a roulette wheel