29a-Integers - Finite Probability (cntd) Introduction...

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Finite Probability (cntd) Discrete Mathematics
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Discrete Mathematics - Integers 29-2 Previous Lecture Experiment, outcomes, sample space Events Classic probability Likelihood of outcomes General probability
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Discrete Mathematics – Finite Probability 29-3 More General Probability Sample space: Any set S Event: `Any’ subset of S Probability: A measure, that is a function Pr: P(S) [0,1], such that - Pr( ) = 0 - Pr(S) = 1 - Pr(A) 0 for all A S - for any disjoint A,B S, Pr(A B) = Pr(A) + Pr(B)
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Discrete Mathematics - Integers 29-4 Properties of Probability Theorem Let S be the sample space of a certain experiment, A,B events. Then a) Pr( ) = 1 – Pr(A) b) Pr(A B) = Pr(A) + Pr(B) – Pr(A B) c) Pr(A B) = Pr(A) · Pr(B) if A and B are `independent’ Proof b) Pr(A B) = Pr(A – B) + Pr(B – A) + Pr(A B) (as these sets are disjoint) = (Pr(A – B) + Pr(A B)) + (Pr(B – A) + Pr(A B)) – Pr(A B) = Pr(A) + Pr(B) – Pr(A B) Q. E. D.
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Discrete Mathematics - Integers 29-5 Examples Two integers are selected, at random and without replacement, from {1,2,…,100}. What is the probability the integers are consecutive?
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29a-Integers - Finite Probability (cntd) Introduction...

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