Fundamental Probability Axioms

Understanding the fundamental probability axioms is key to grasping how we measure uncertainty. These principles, like sample spaces and probability measures, lay the groundwork for calculating event likelihoods and making informed decisions based on data.

1. Sample space and events

   • The sample space (S) is the set of all possible outcomes of a random experiment.
   • Events are subsets of the sample space, representing specific outcomes or groups of outcomes.
   • Events can be simple (single outcome) or compound (multiple outcomes).
   • Understanding the sample space is crucial for defining probabilities of events.

2. Probability measure

   • A probability measure assigns a numerical value to events, indicating the likelihood of their occurrence.
   • It is a function P that maps events to the interval [0, 1].
   • The probability measure must satisfy the axioms of probability, ensuring consistency and validity.

3. Non-negativity axiom

   • The non-negativity axiom states that the probability of any event is always greater than or equal to zero.
   • Mathematically, for any event A, P(A) ≥ 0.
   • This axiom ensures that probabilities are meaningful and cannot be negative.

4. Unity axiom

   • The unity axiom states that the probability of the entire sample space is equal to one.
   • Mathematically, P(S) = 1, where S is the sample space.
   • This axiom establishes a baseline for measuring probabilities of individual events.

5. Additivity axiom

   • The additivity axiom states that for mutually exclusive events, the probability of their union is the sum of their individual probabilities.
   • Mathematically, if A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B).
   • This axiom allows for the calculation of probabilities for combined events.

6. Complement rule

   • The complement rule states that the probability of an event not occurring is equal to one minus the probability of the event occurring.
   • Mathematically, P(A') = 1 - P(A), where A' is the complement of event A.
   • This rule is useful for calculating probabilities when direct computation is difficult.

7. Inclusion-exclusion principle

   • The inclusion-exclusion principle provides a way to calculate the probability of the union of multiple events.
   • It accounts for overlapping events to avoid double counting.
   • Mathematically, for two events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

8. Conditional probability

   • Conditional probability measures the probability of an event occurring given that another event has already occurred.
   • Mathematically, P(A | B) = P(A ∩ B) / P(B), where P(B) > 0.
   • This concept is essential for understanding dependencies between events.

9. Law of total probability

   • The law of total probability relates the probability of an event to a partition of the sample space.
   • It states that if B1, B2, ..., Bn are mutually exclusive and exhaustive events, then P(A) = Σ P(A | Bi)P(Bi).
   • This law is useful for breaking down complex probability calculations.

10. Bayes' theorem

• Bayes' theorem provides a way to update probabilities based on new evidence.
• Mathematically, P(A | B) = [P(B | A)P(A)] / P(B).
• This theorem is fundamental in statistical inference and decision-making under uncertainty.