Lecture Notes on Probability Concepts

Axioms of Probability

• Definition: Axioms are the fundamental assumptions of probability.

Axiom 1

• Statement: For any event, the probability of that event is greater than or equal to zero.
  • Clarification: The probability must also be less than or equal to one.

Axiom 2

• Statement: The probability of the sample space is equal to one.
  • Sample Space: The set of all possible outcomes of an experiment.

Axiom 3

• Statement: If there are infinite disjoint events (events whose intersection is null), then the probability of their union is the sum of their individual probabilities.
  • Notation: For events A1, A2,..., An, the probability of A1 ∪ A2 ∪ ... = P(A1) + P(A2) + ...

Key Definitions

• Event: Any subset of the sample space.
• Disjoint Events: Events that do not share any outcomes (intersection is null).

Example of Axiom 3

• Experiment: Throwing a die.
  • Events:
    • A1: getting 1
    • A2: getting 2
    • A3: getting 3
    • A4: getting 4
    • A5: getting 5
    • A6: getting 6

Proposed Proofs

Proposition 1: Probability of Null Event is Zero

• Goal: Prove that P(∅) = 0.
• Method: Using Axiom 3 with events that have no members.
  • Example: Consider events A1, A2... where each event represents outcomes that cannot occur (e.g., rolling a 7 with a die).
  • All events being disjoint leads to P(∅) = P(A1) + P(A2) + ... = 0.

Proposition 2: Finite Collection of Disjoint Events

• Goal: Prove that for k disjoint events A1, A2,..., Ak, the probability of their union equals the sum of their probabilities.
  • Statement: P(A1 ∪ A2 ∪ ... ∪ Ak) = P(A1) + P(A2) + ... + P(Ak)
• Method: Augment the set with artificial null events to apply Axiom 3.
  • Use the fact that the sum of probabilities of null events is zero.

Conclusion

• The axioms provide foundational rules for calculating probabilities.
• The derived propositions show how to compute probabilities for disjoint events both in finite and infinite cases.