APME_1 - Applied Probability Methods for Engineers Slide...

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Applied Probability Methods for Engineers Slide Set 1
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Chapter 1 Probability Theory
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Probability and Statistics How would we define the word probability ? Likelihood, relative frequency, chance How would we define the word statistic ? A property or measure associated with a sample taken from a population, typically used to make inferences about the population Probability and statistics tools have been developed to help us deal with things we do not or can not know with certainty
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Introductory Probability Theory Experiment: process that can lead to more than one outcome Sample Space: Set of all possible outcomes associated with an experiment Sample Space for picking a card from a deck:
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Introductory Probability Theory Sample space for choosing two cards with replacement:
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Introductory Probability Theory Sample space for choosing two cards without replacement:
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Probability Values Given a sample space S consisting of n outcomes { O 1 , O 2 , . .., O n }, a set of n probability values satisfies: 0 ≤ p 1 ≤ 1, 0 ≤ p 2 ≤ 1, …, 0 ≤ p n ≤ 1, and p 1 + p 2 + … + p n = 1 The probability of outcome O i occurring is p i , and we write this as P( O i ) = p i
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Probability Values Associated with picking a single card:
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Probability Values Associated with picking two cards with replacement
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Probability Values Associated with picking two cards without replacement:
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Events/Complements An event A is a subset of sample space S, and the event A occurs if one of the outcomes contained in the event occurs P(A) denotes the probability of event A, and is obtained by summing probabilities of the outcomes associated with the event P(A) = 0.10 + 0.15 + 0.30 = 0.55
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Events/Complements The event A' is everything not contained in A, and is called the complement of A P(A') = 0.10 + 0.05 + 0.05 + 0.15 + 0.10 = 0.45 = 1 – P(A)
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Event Examples Probability a randomly drawn cards belongs to hearts suit (event A) P(A) = 13*(1/52) = 13/52 = 1/4
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Event Examples Probability a randomly selected card is a picture card (event B) P(B) = 12*(1/52) =12/52 = 3/13
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Event Combinations Intersection of events A and B, denoted A ∩ B, consists of all outcomes in both A and B P(A ∩ B) is the probability both A and B occur at the same time Clearly P(A ∩ A') = P( ) = 0 P(A ∩ B) + P(A ∩ B') = P(A) P(A ∩ B) + P(A' ∩ B) = P(B) Events A and B are mutually exclusive if they have no common outcomes In this case A ∩ B = and P(A ∩ B) = 0
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The union of two events A and B, denoted by A B, consists of all outcomes either in A or B or both P(A B) is the probability that at least one of the events A, B occurs P(A B) = P(A ∩ B') + P(A' ∩ B) + P(A ∩ B) P(A ∩ B') = P(A) – P(A ∩ B) and P(A' ∩ B) = P(B) – P(A ∩ B), implying P(A B) = P(A) + P(B) – P(A ∩ B) When A and B are mutually exclusive, then P(A ∩ B) =
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APME_1 - Applied Probability Methods for Engineers Slide...

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