2.2 Properties of probability

Probability properties form the backbone of understanding chance and uncertainty in various scenarios. These fundamental concepts, including non-negativity, boundedness, and normalization, provide a framework for quantifying likelihood and making informed decisions.

From weather forecasting to medical diagnoses, probability properties find practical applications across diverse fields. By applying these principles, we can assess risks, predict outcomes, and develop strategies in areas ranging from finance and engineering to sports and marketing.

Properties of Probability

Fundamental Concepts of Probability

• Probability measures likelihood of event occurrence expressed as number between 0 and 1
• Non-negativity property requires probability of any event to be greater than or equal to zero
• Boundedness constrains probabilities to be less than or equal to 1
• Normalization property dictates sum of probabilities for all possible outcomes in sample space equals 1
• Probability of empty set (impossible event) always 0
• Probability of entire sample space (certain event) always 1
• Complementary events have probabilities summing to 1
  • Reflects fact that either event or its complement must occur
  • Example: Probability of rolling even number on die (P(even) = 0.5) and probability of rolling odd number (P(odd) = 0.5) sum to 1
• Monotonicity property states if event A is subset of event B, probability of A less than or equal to probability of B
  • Example: Probability of rolling a 6 on a die (P(6) = 1/6) is less than probability of rolling an even number (P(even) = 1/2)

Applications of Probability Properties

• Non-negativity and boundedness ensure meaningful risk probabilities in assessment scenarios
  • Example: Probability of flight delay cannot be negative or exceed 100%
• Additivity property crucial for calculating overall probability of complex events
  • Used in engineering reliability and financial risk management
  • Example: Probability of system failure calculated by adding probabilities of individual component failures
• Weather forecasting utilizes normalization property
  • Ensures probabilities of all possible weather outcomes sum to 1
  • Example: Probabilities of sunny (60%), cloudy (30%), and rainy (10%) weather sum to 100%
• Medical diagnosis applies complement rule
  • Relates probabilities of having and not having particular condition
  • Example: If probability of having flu is 0.2, probability of not having flu is 0.8
• Gambling and game theory rely on probability properties for odds calculation and strategy development
  • Example: Calculating probability of winning in poker based on hand combinations
• Quality control uses monotonicity property to compare defect probabilities
  • Compares different manufacturing processes or batches
  • Example: Probability of defect in newer manufacturing process (2%) less than in older process (5%)
• Insurance companies model and price risk scenarios using probability properties
  • Ensures actuarially fair premiums and adequate reserves
  • Example: Calculating probability of car accident to determine insurance premium

Proof of Probability Axioms

Kolmogorov's Axioms and Basic Proofs

• Three fundamental axioms of probability defined by Kolmogorov form foundation for deriving all other probability properties
• Axiom 1: Probability of any event is non-negative
  • Expressed as $P(A) \geq 0$ for any event A in sample space
• Axiom 2: Probability of entire sample space is 1
  • Expressed as $P(S) = 1$, where S is sample space
• Axiom 3 (Additivity axiom): For mutually exclusive events A and B, $P(A \cup B) = P(A) + P(B)$
• Complement rule proven using axioms: $P(A') = 1 - P(A)$, where A' is complement of event A
  • Proof:
    1. $P(A) + P(A') = P(S) = 1$ (Axiom 2 and Axiom 3)
    2. $P(A') = 1 - P(A)$ (Rearranging the equation)
• Inclusion-exclusion principle for two events derived from axioms
  • Expressed as $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • Proof:
    1. $P(A \cup B) = P(A) + P(B \setminus A)$ (Axiom 3)
    2. $P(B \setminus A) = P(B) - P(A \cap B)$ (Complement rule)
    3. Substituting (2) into (1) gives the desired result

Advanced Proofs and Theorems

• Law of total probability proven using axioms and derived properties
  • Expressed as $P(A) = \sum_{i} P(A|B_i)P(B_i)$, where $B_i$ form partition of sample space
  • Proof involves applying additivity axiom and definition of conditional probability
• Bayes' theorem constructed using axioms and derived properties
  • Expressed as $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
  • Proof:
    1. $P(A \cap B) = P(A|B)P(B)$ (Definition of conditional probability)
    2. $P(A \cap B) = P(B|A)P(A)$ (Same as step 1, but swapping A and B)
    3. Equating (1) and (2) and rearranging gives Bayes' theorem
• Monotonicity property proven using axioms
  • If A is subset of B, then $P(A) \leq P(B)$
  • Proof:
    1. $B = A \cup (B \setminus A)$ (Set theory)
    2. $P(B) = P(A) + P(B \setminus A)$ (Axiom 3)
    3. $P(B \setminus A) \geq 0$ (Axiom 1)
    4. Therefore, $P(B) \geq P(A)$

Probability in Real-World Scenarios

Risk Assessment and Decision Making

• Non-negativity and boundedness properties ensure meaningful risk probabilities
  • Example: Probability of stock market crash cannot be negative or exceed 100%
• Additivity property crucial for calculating overall probability of complex events
  • Used in engineering reliability and financial risk management
  • Example: Probability of successful product launch calculated by combining probabilities of market acceptance, production efficiency, and distribution success
• Complement rule applied in medical diagnosis
  • Relates probabilities of having and not having particular condition
  • Example: If probability of having cancer is 0.05, probability of not having cancer is 0.95
• Insurance companies use probability properties to model and price risk scenarios
  • Ensures actuarially fair premiums and adequate reserves
  • Example: Calculating probability of natural disasters to determine homeowners insurance rates

Scientific and Industrial Applications

• Weather forecasting relies on normalization property
  • Ensures probabilities of all possible weather outcomes sum to 1
  • Example: Probabilities of clear skies (50%), partly cloudy (30%), and overcast (20%) sum to 100%
• Quality control uses monotonicity property to compare defect probabilities
  • Compares different manufacturing processes or batches
  • Example: Probability of defect in automated assembly line (1.5%) less than in manual assembly (3%)
• Particle physics experiments apply probability properties to analyze collision data
  • Example: Calculating probability of observing Higgs boson in Large Hadron Collider data
• Environmental science uses probability to model climate change scenarios
  • Example: Probability of sea level rise exceeding certain threshold under different emission scenarios

Games and Strategic Decision Making

• Gambling and game theory rely on probability properties for odds calculation and strategy development
  • Example: Calculating probability of winning in blackjack based on visible cards and dealer's up card
• Sports analytics apply probability concepts to optimize team strategies
  • Example: Probability of successful three-point shot versus two-point shot in basketball
• Political polling uses probability sampling to predict election outcomes
  • Example: Calculating margin of error in voter preference surveys based on sample size and population characteristics
• Marketing strategies employ probability models to predict consumer behavior
  • Example: Probability of customer purchasing product based on demographic factors and previous buying patterns