GE 331-Lecture 4

GE 331-Lecture 4 - Axioms of probability Axioms P()=1...

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IE 300/GE 331 Lecture 4 Negar Kiyavash, UIUC 1 Axioms of probability Axioms P( )=1 0 P(A) 1 for any event A For mutually exclusive events A and B, P(AUB)=P(A)+P(B) Properties: P(A')=1-P(A): P(A)+P(A')=P(AUA')=P( )=1 P(Ø)=0
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IE 300/GE 331 Lecture 4 Negar Kiyavash, UIUC 2 Addition rules Think of probability as area Area(AUB) = Area(A) + Area(B) – Area(A B) Addition rule P(AUB)=P(A)+P(B)-P(A B)
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IE 300/GE 331 Lecture 4 Negar Kiyavash, UIUC 3 Partitions For any event A єΩ : A U A’= and A A’=Ø. A and A’ are a partition of . A B and A B’ are partitions of A.
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IE 300/GE 331 Lecture 4 Negar Kiyavash, UIUC 4 Conditional probability (cont) The conditional probability of an event B given an event A is P(B|A) = P(A B)/P(A) where P(A) is not zero The probability of attending college is higher for someone whose parents attended college than for someone whose parents did not. In decision analysis, AI and machine learning, computational finance, communication and information theory: likelihood functions. These figure in both Bayesian and classical versions of statistical inference
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IE 300/GE 331 Lecture 4 Negar Kiyavash, UIUC 5 Conditional probability (cont) Example: Two exams are given in a class. A student passes the 1 st exam with probability 0.8, and passes both exams with probability 0.72. What is the probability that a student who passes the 1 st exam also passes the 2 nd exam?
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Negar Kiyavash, UIUC 6 Conditional probability (cont) A: one passes the 1 st exam, B: one passes the 2 nd exam P(B|A)? P(A)=0.8, P(A
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GE 331-Lecture 4 - Axioms of probability Axioms P()=1...

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