2.1 Axioms of probability

Probability axioms are the foundation of probability theory, providing a consistent framework for calculating and interpreting probabilities. These axioms ensure that probabilities are non-negative, the total probability of all possible outcomes is 1, and mutually exclusive events can be added together.

Understanding these axioms is crucial for solving probability problems and avoiding common pitfalls. They lead to important concepts like the complement rule, inclusion-exclusion principle, and conditional probability, which are essential tools for tackling more complex probability scenarios in real-world applications.

Axioms of Probability

Fundamental Axioms

Top images from around the web for Fundamental Axioms

• 

  Introduction to Probability Rules | Concepts in Statistics View original

  Is this image relevant?

• 

  Probability axioms - Wikipedia View original

  Is this image relevant?

• 

  Tree diagram (probability theory) - Wikipedia View original

  Is this image relevant?

• 

  Introduction to Probability Rules | Concepts in Statistics View original

  Is this image relevant?

• 

  Probability axioms - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamental Axioms

• 

  Introduction to Probability Rules | Concepts in Statistics View original

  Is this image relevant?

• 

  Probability axioms - Wikipedia View original

  Is this image relevant?

• 

  Tree diagram (probability theory) - Wikipedia View original

  Is this image relevant?

• 

  Introduction to Probability Rules | Concepts in Statistics View original

  Is this image relevant?

• 

  Probability axioms - Wikipedia View original

  Is this image relevant?

1 of 3

• First axiom states probability of any event A is non-negative real number $[P(A)](https://www.fiveableKeyTerm:p(a)) \geq 0$ for all A in sample space S
• Second axiom states probability of entire sample space S equals 1 $P(S) = 1$
• Third axiom (countable additivity) states for mutually exclusive events, probability of their union equals sum of individual probabilities
• Axioms form foundation of probability theory ensuring consistent and well-defined probability measures

Implications of Axioms

• Probability of any event always between 0 and 1, inclusive $0 \leq P(A) \leq 1$ for all A in S
• Probability of complement of event A equals 1 minus probability of A $P(A') = 1 - P(A)$
• Axioms lead to inclusion-exclusion principle for calculating probability of union of non-mutually exclusive events
• Enable calculation of compound event probabilities (flipping coin twice)
• Allow for normalization of probability distributions (ensuring probabilities sum to 1)

Applying Probability Axioms

Basic Problem-Solving Techniques

• Utilize first axiom to ensure calculated probabilities are non-negative for single events or combinations
• Apply second axiom to normalize probability distributions and verify sum of probabilities equals 1 in discrete problems
• Implement third axiom to calculate compound event probabilities by adding individual probabilities of mutually exclusive events
• Use complement rule to simplify complex calculations involving complementary events
• Apply inclusion-exclusion principle for problems with non-mutually exclusive events
  • Example: Probability of selecting a red or face card from a standard deck
• Utilize conditional probability concept for problems with dependent events
  • Example: Probability of drawing two aces without replacement

Advanced Applications

• Recognize and apply multiplication rule for independent events in problems with multiple trials
  • Example: Probability of rolling a 6 three times in a row with a fair die
• Use axioms to derive and apply Bayes' theorem for updating probabilities based on new information
• Apply axioms in continuous probability distributions (integrating probability density functions)
• Utilize axioms in solving problems involving expected values and variances
• Apply axioms in more complex scenarios like Markov chains or Poisson processes

Violations of Probability Axioms

Common Violations

• Identify scenarios with negative probabilities, violating first axiom of non-negativity
  • Example: Incorrectly calculating probability of an impossible event as -0.2
• Detect cases where sum of probabilities for all outcomes exceeds or falls short of 1, violating second axiom
  • Example: Assigning probabilities of 0.3, 0.4, and 0.5 to three mutually exclusive and exhaustive events
• Recognize situations where probability of mutually exclusive events not equal to sum of individual probabilities, violating third axiom
  • Example: Claiming P(A or B) = P(A) × P(B) for mutually exclusive events A and B

Inconsistencies and Paradoxes

• Identify improper probability assignments leading to inconsistencies or paradoxes
  • Example: Bertrand's paradox, where different methods of choosing random chords on a circle yield different probabilities
• Detect instances where complement rule not satisfied, indicating violation of axioms' implications
  • Example: Claiming P(A) = 0.7 and P(not A) = 0.4
• Recognize scenarios where inclusion-exclusion principle fails, suggesting violation of fundamental axioms
  • Example: Incorrectly applying the principle to dependent events
• Identify cases where conditional probabilities inconsistent with axioms
  • Example: Assigning P(A|B) > 1 or P(A|B) + P(not A|B) ≠ 1
• Recognize violations in more advanced probability concepts (Kolmogorov's zero-one law, martingales)