Properties of Probability

Review of Set Theory SHIWEI LAN

terminology a: outcome, result of a random experiment A: event, set of outcomes S: outcome space, collection of all possible outcomes We have 𝑎∈A⊂𝑆

Multiple Events A,B,𝐶⊂𝑆 Subset: A⊂𝐵 Union: A∪𝐵 Intersection: A∩𝐵 If A∩𝐵=∅, then A, B are mutually exclusive events.

Properties Communtative: A∪𝐵=𝐵∪𝐴, A∩𝐵=𝐵∩𝐴 Associative: (A∪𝐵)∪𝐶=A∪(𝐵∪𝐶), (A∩𝐵)∩𝐶=A∩ (𝐵∩𝐶) Distributive: A∪(𝐵∩𝐶)= A∪𝐵 ∩ A∪𝐶 , A∩(𝐵∪𝐶)= (A∩𝐵)∪ (A∩𝐶) De Morgan’s Laws: A∪𝐵 ′ = A′∩𝐵′, (A∩𝐵)′= A′∪𝐵′

Probability Probability is a real-valued set function P that assigns, to each event A in the sample space S, a number P(A), called the probability of the event A, such that the following properties are satisfied: i). P A ≥0 ii). P S =1 iii). If { 𝐴 𝑖 :𝑖=1,…,𝐿} are mutually exclusive, then P ∪ 𝑖=1 𝐿 𝐴 𝑖 = 𝑖=1 𝐿 𝑃( 𝐴 𝑖 ) , for 𝐿 is finite or ∞

Theorems Thm1. P A =1−P A′ Thm2. P ∅ =0 Thm3. if A⊂𝐵, then P A ≤P B . Show 𝐵=𝐴∪(𝐵 ∩𝐴 ′ ) and 𝐴∩ 𝐵 ∩𝐴 ′ =∅ Thm4. P A ≤1 Thm5. P 𝐴∪𝐵 =P A +P B −P 𝐴∩𝐵 . Show𝐴∪𝐵= 𝐴∪( 𝐴 ′ ∩ 𝐵) , and B=(𝐴∩𝐵)∪(𝐴′∩𝐵) Thm6. P 𝐴∪𝐵∪𝐶 =P A +P B +P C −P 𝐴∩𝐵 −P 𝐵∩𝐶 𝐵∩𝐶 −P 𝐴∩C +P 𝐴∩𝐵∩C . Use Thm5.

Examples 1. A survey was taken of a group’s viewing habits of sporting events on TV during the last year. Let A = {watched football}, B = {watched basketball}, C = {watched baseball}. The results indicate that if a person is selected at random from the surveyed group, then P(A) = 0.43, P(B) = 0.40, P(C) = 0.32, P(A ∩ B) = 0.29, P(A∩C) = 0.22, P(B∩C) = 0.20, and P(A∩B∩C) = 0.15. What is the probability that this person watched at least one of these sports? (A) 0.5 (B) 0 (C) 1 (D) 0.59

Examples 2. The probability that a randomly selected student at Anytown College owns a bicycle is 0.55, the probability that a student owns a car is 0.30, and the probability that a student owns both is 0.10. What is the probability that a student selected at random does not own a bicycle? (A) 0.5 (B) 0.45 What is the probability that a student selected at random owns at least a car or a bicycle? (A) 0.85 (B) 0.75 What is the probability that a student selected at random has neither a car nor a bicycle? (A) 0.15 (B) 0.25