Probability & Statistics, Lecture 3A: Laws of Probability (Axioms, Theorems, Proofs)

Calculus-based Probability and Statistics for Engineers and Scientists, Lecture 3, Part 1. Also for Data Scientists and Actuarial Science majors.

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(0:00) Venn diagrams will be useful for understanding and problem solving
(0:59) Probability Axiom 1: P(S) = 1 where S is the sample space
(3:49) Probability Axiom 2: P(A) ≥ 0 for any event A
(5:46) Probability Axiom 3: P(A ∪ B) = P(A) + P(B) when A and B are mutually exclusive (Addition Rule for mutually exclusive events)
(11:43) Probability Theorem 1: P(∅) = 0 where ∅ is the empty set (also written ∅ = { })
(13:05) ∅ is a subset of any set (this is vacuously true)
(13:34) Subset definition
(14:59) Truth table for implication P implies Q
(17:06) Probability Theorem 2: P(A') = 1 - P(A), where A' is the set complement of A in S
(18:39) Definition of set complement
(20:26) Probability Theorem 3: General Addition Rule P(A ∪ B) = P(A) + P(B) - P(A ⋂ B)
(22:41) Venn Diagram for General Addition Rule
(25:28) Axioms should be minimal. Theorems require proof.
(26:32) Proof of Theorem 1 (law 4)
(36:22) Sketch of proof of Theorem 2 (law 5)
(39:15) Homework guidance to prove: If A is a subset of B, then P(A) ≤ P(B)

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