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Probability

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The calculated likelihood that a given event will occur

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Methods of Determining Probability Empirical Experimental observation Example – Process control Theoretical Uses known elements Example – Coin toss, die rolling Subjective Assumptions Example – I think that...

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Probability Components Experiment An activity with observable results Sample Space A set of all possible outcomes Event A subset of a sample space Outcome / Sample Point The result of an experiment

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Probability What is the probability of a tossed coin landing heads up? Probability Tree Experiment Sample Space Event Outcome

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Probability A way of communicating the belief that an event will occur. Expressed as a number between 0 and 1 fraction, percent, decimal, odds Total probability of all possible events totals 1

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Relative Frequency The number of times an event will occur divided by the number of opportunities = Relative frequency of outcome x = Number of events with outcome x n = Total number of events Expressed as a number between 0 and 1 fraction, percent, decimal, odds Total frequency of all possible events totals 1

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Probability What is the probability of a tossed coin landing heads up? How many possible outcomes? 2 How many desirable outcomes? 1 Probability Tree What is the probability of the coin landing tails up?

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Probability How many possible outcomes? How many desirable outcomes? 1 What is the probability of tossing a coin twice and it landing heads up both times? 4 HH HT TH TT

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Probability How many possible outcomes? How many desirable outcomes? 3 What is the probability of tossing a coin three times and it landing heads up exactly two times? 8 1 st 2 nd 3 rd HHH HHT HTH HTT THH THT TTH TTT

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Binomial Process Each trial has only two possible outcomes yes-no, on-off, right-wrong Trial outcomes are independent Tossing a coin does not affect future tosses

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Bernoulli Process P = Probability x = Number of times for a specific outcome within n trials n = Number of trials p = Probability of success on a single trial q = Probability of failure on a single trial ! = factorial – product of all integers less than or equal

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Probability Distribution What is the probability of tossing a coin three times and it landing heads up two times?

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Law of Large Numbers Trial 1: Toss a single coin 5 times H,T,H,H,T P =.600 = 60% Trial 2: Toss a single coin 500 times H,H,H,T,T,H,T,T,……T P =.502 = 50.2% Theoretical Probability =.5 = 50% The more trials that are conducted, the closer the results become to the theoretical probability

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Probability Independent events occurring simultaneously Product of individual probabilities If events A and B are independent, then the probability of A and B occurring is: P(A and B) = P AP B AND (Multiplication)

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Probability AND (Multiplication) What is the probability of rolling a 4 on a single die? How many possible outcomes? How many desirable outcomes? 1 6 What is the probability of rolling a 1 on a single die? How many possible outcomes? How many desirable outcomes? 1 6 What is the probability of rolling a 4 and then a 1 in sequential rolls?

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Probability Independent events occurring individually Sum of individual probabilities If events A and B are mutually exclusive, then the probability of A or B occurring is: P(A or B) = P A + P B OR (Addition)

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Probability OR (Addition) What is the probability of rolling a 4 on a single die? How many possible outcomes? How many desirable outcomes? 1 6 What is the probability of rolling a 1 on a single die? How many possible outcomes? How many desirable outcomes? 1 6 What is the probability of rolling a 4 or a 1 on a single die?

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Probability Independent event not occurring 1 minus the probability of occurrence P = 1 - P(A) NOT What is the probability of not rolling a 1 on a die?

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How many tens are in a deck? Probability Two cards are dealt from a shuffled deck. What is the probability that the first card is an ace and the second card is a face card or a ten? How many cards are in a deck? 52 4 12 4 How many aces are in a deck? How many face cards are in deck?

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Probability What is the probability that the first card is an ace? Since the first card was NOT a face, what is the probability that the second card is a face card? Since the first card was NOT a ten, what is the probability that the second card is a ten?

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Probability Two cards are dealt from a shuffled deck. What is the probability that the first card is an ace and the second card is a face card or a ten? If the first card is an ace, what is the probability that the second card is a face card or a ten? 31.37%

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Conditional Probability P(E|A) = Probability of event E, given A Example: One card is drawn from a shuffled deck. The probability it is a queen is P(queen) = However, if I already know it is face card P(queen | face)=

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Conditional Probability Probability of two events A and B both occurring = P(A and B) = P(A|B) P(B) = P(B|A) P(A) If A and B are independent, then P(A and B) = P(A) P(B)

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Bayes Theorem Calculates a conditional probability, based on all the ways the condition might have occurred. P( A | E ) = probability of A, given we already know the condition E =

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Bayes Theorem Example LCD screen components for a large cell phone manufacturing company are outsourced to three different vendors. Vendor A, B, and C supply 60%, 30%, and 10% of the required LCD screen components. Quality control experts have determined that.7% of vendor A, 1.4% of vendor B, and 1.9% of vendor C components are defective. If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor A?

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Bayes Theorem Example Notation Used: P = Probability D = Defective A, B, and C denote vendors Unknown to be calculated: Probability the screen is from A, given that it is defective

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Bayes Theorem Example Known probabilities: Probability the screen is from A Probability the screen is from B Probability the screen is from C

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Bayes Theorem Example Probability the screen is defective given it is from C Probability the screen is defective given it is from A Probability the screen is defective given it is from B Known conditional probabilities:

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Bayes Theorem Example: Defective Part = P(screen is defective AND from A) P(screen is defective from anywhere)

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LCD Screen Example

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If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor B? If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor C?

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