APME_1 - Click to edit Master subtitle style Applied...

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

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