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AxiomsOfProbability

# AxiomsOfProbability - • If the coin and dice are both...

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Learning Objectives State the three axioms of probability.

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Example After an exam, a doctor concludes that a patient has either plague, leprosy, or SARS. She decides that the probability is 0.7 that the patient has plague, 0.2 that the patient has leprosy, and 0.1 that the patient has SARS. What is the probability that the patient has either plague or leprosy?
Axioms of Probability Non-negativity Additivity Normalization

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Non-negativity For every event A, P ( A ) ³ 0
Additivity The probability of the union of disjoint events is the sum of their individual probabilities P ( A 1 È A 2 È ...) = P ( A 1 ) + P ( A 2 ) +. ..

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Normalization The probability of the entire sample space ( ) is equal to one P (W) =1
Example Experiment: flip a coin and roll a six-sided dice Sample space (outcomes): ={H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}

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Unformatted text preview: • If the coin and dice are both “fair”, then each of the 12 outcomes is equally likely. • Additivity & normalization -> P({})=1/12 Example • Calculate probabilities of events by counting the number of outcomes and dividing by the total number of outcomes • P(A) = 6/12 • P(B) = 6/12 • P(C) = 2/12 A = H 1, H 2, H 3, H 4, H 5, H 6, { } B = H 1, H 3, H 5, T 1, T 3, T 5 { } C = H 3, T 3 { } Does this conflict with the additivity axiom? More Probability Properties P(Ø) = 0 P(A c ) = 1 - P(A) If A ⊂ B then P(A) ≤ P(B) P(B) = P(A ∩ B) + P(A c ∩ B) P(A ∪ B) = P(A) + P(A c ∩ B) P(A ∪ B) = P(A) + P(B) - P(A ∩ B) P(A ∪ B ∪ C) = P(A) + P(A c ∩ B) + P(A c ∩ B c ∩ C) Exercises • Exercise #1 • Exercise #2 • Exercise #3...
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AxiomsOfProbability - • If the coin and dice are both...

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