Tutorial 2:Sample Space and Probability 2 Baoxiang WANG bxwang@cse Spring 2017
Tutorial 2: Sample Space and Probability 2 Baoxiang WANG bxwang@cse Spring 2017
Probability models A probability model is an assignment of probabilities to every element of the sample space. Probabilities are nonnegative and add up to one. Examples S=HH,HT,TH,TT /4/44/4 models a pair of coins with equally likely outcomes
Probability models A probability model is an assignment of probabilities to every element of the sample space. Probabilities are nonnegative and add up to one. S = { HH, HT, TH, TT } ¼ ¼ ¼ ¼ Examples models a pair of coins with equally likely outcomes
Elements of a Probabilistic Model Event:a subset of sample space. -A is a set of possible outcomes -Example.A ={HH,TT},the event that the two coins give the same side. The probability law assigns our knowledge or belief to an event Aa number P(A)>0. It specifies the likelihood of any outcome
Elements of a Probabilistic Model • Event: a subset of sample space. – 𝐴 ⊆ Ω is a set of possible outcomes – Example. 𝐴 = 𝐻𝐻, 𝑇𝑇 , the event that the two coins give the same side. • The probability law assigns our knowledge or belief to an event 𝐴 a number 𝑃 𝐴 ≥ 0. – It specifies the likelihood of any outcome
Probability Axioms 1.(Non-negativity)P(A)>0,for every event A. 2.(Additivity)For any two disjoint events A and B,P(AUB)=P(A)+P(B) In general,if A1,A2,..are disjoint events, then P(A1UA2U…)=P(A1)+P(A2)+… 3.(Normalization)P()=1
Probability Axioms 1. (Non-negativity) 𝑃(𝐴) ≥ 0, for every event 𝐴. 2. (Additivity) For any two disjoint events 𝐴 and 𝐵, 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 In general, if 𝐴1 , 𝐴2 , … are disjoint events, then 𝑃 𝐴1 ∪ 𝐴2 ∪ ⋯ = 𝑃 𝐴1 + 𝑃 𝐴2 + ⋯ 3. (Normalization) 𝑃(Ω) = 1
De Morgan's laws AUB)C=AcnBC (AnB)=ACUBC U U B A B' A'uB' (A1UUAn)c=An…nA9 (A1n…nAn)c=AUUA
De Morgan’s laws 𝐴 ∪ 𝐵 c = A 𝑐 ∩ 𝐵 𝑐 𝐴 ∩ 𝐵 c = A 𝑐 ∪ 𝐵 𝑐 𝐴1 ∪ ⋯ ∪ 𝐴𝑛 c = 𝐴1 𝑐 ∩ ⋯ ∩ 𝐴𝑛 𝑐 𝐴1 ∩ ⋯ ∩ 𝐴𝑛 c = 𝐴1 𝑐 ∪ ⋯ ∪ 𝐴𝑛 𝑐