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Jan. 14 1.2 Properties of Probability Event: A subset of the sample space (technical restrictions unimportant for now) Often use early-alphabet capital letters (e.g., A, B, C) for events. Difference between an outcome and an event? After an experiment, event A has occurred if . . . Jan. 14 1.2 Properties of Probability Algebra of sets and Venn diagrams Null event and full sample space: ∅ and S Subset notation: A ⊂ B or, alternatively, B ⊃ A or A implies B Union: A ∪ B Intersection: A ∩ B Complement: A0 or, alternatively, Ac Special random variable: The indicator IA or I(A) or 1A Jan. 14 1.2 Properties of Probability Mutually exclusive events: Have empty pairwise intersection Exhaustive events: Have union equal to S Partition: A group of mutually exclusive and exhaustive events Jan. 14 1.2 Properties of Probability An example: Roll two dice, so S has 36 elements or outcomes. Let A = {s ∈ S : sum of dice results in s is even} B = {first die is even} (using abbreviated notation) C = {second die is ≥ 5} D = {sum is prime} What is A ∩ D? (B ∪ C 0 ∪ A)0 ? (A0 ∩ B) ∪ (A ∩ B)? Jan. 14 Appendix A.1 Algebra of sets Jan. 14 Commutative property: A ∪ B = B ∪ A Appendix A.1 Algebra of sets Commutative property: A ∩ B = B ∩ A Associative property: (A ∪ B) ∪ C = A ∪ (B ∪ C) Associative property: (A ∩ B) ∩ C = A ∩ (B ∩ C) Distributive property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Distributive property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) DeMorgan’s Law: (A ∪ B)0 = A0 ∩ B 0 or, more generally, DeMorgan’s Law: (A ∩ B)0 = A0 ∪ B 0 or, more generally, n [ i=1 Jan. 14 !0 Ai = n \ i=1 1.2 Properties of Probability Probability: Real-valued set function, P, satisfying: P(A) ≥ 0 (Nonnegativity) If A is the whole sample space S then P(A) = 1 Whenever A1 , A2 , . . . are mutually exclusive, ! ∞ ∞ [ X P Ai = P(Ai ) (Countable additivity) i=1 i=1 n \ A0i !0 Ai i=1 Jan. 14 1.2 = n [ A0i i=1 Properties of Probability Some theorems: (1) For any A, P(A) = 1 − P(A0 ). (2) P(∅) = 0. (3) If A implies B (i.e., A ⊂ B), then P(A) ≤ P(B). (4) For any A, P(A) ≤ 1. (5) For any A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (5∗ ) For any A and B and C, P(A ∪ B ∪ C) = P(A) + P(B) + P(C) −P(A ∩ B) − P(A ∩ C) − P(B ∩ C) +P(A ∩ B ∩ C) (5∗∗ ) For any A1 , . . . , An , ! n n XX [ X P Ai = P(Ai ) − P(Ai ∩ Aj ) + · · · i=1 i=1 i<j +(−1)n+1 P(A1 ∩ · · · ∩ An ) Jan. 14 Desired Outcomes Students will be able to: apply set notation to simplify expressions involving events; apply Venn diagrams to the same; prove simple facts based on the three axioms of probability;