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Probability

Probability. Probability. The calculated likelihood that a given event will occur. Methods of Determining Probability. Empirical Experimental observation Example – Process control Theoretical Uses known elements Example – Coin toss, die rolling

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Probability

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  1. Probability

  2. Probability The calculated likelihood that a given event will occur

  3. Methods of Determining Probability • Empirical • Experimental observation Example – Process control • TheoreticalUses known elements • Example – Coin toss, die rolling • SubjectiveAssumptions Example – I think that . . .

  4. ProbabilityComponents • ExperimentAn activity with observable results • Sample SpaceA set of all possible outcomes • EventA subset of a sample space • Outcome / Sample PointThe result of an experiment

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

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

  7. 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 1fraction, percent, decimal, odds Total frequency of all possible events totals 1

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

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

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

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

  12. 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

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

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

  15. Probability AND (Multiplication) 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) = PA∙PB

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

  17. Probability OR (Addition) 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) = PA + PB

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

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

  20. 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? 52 How many cards are in a deck? How many aces are in a deck? 4 12 How many face cards are in deck? How many tens are in a deck? 4

  21. 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?

  22. 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%

  23. 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)=

  24. 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)

  25. 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 =

  26. 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?

  27. 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

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

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

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

  31. LCDScreenExample

  32. LCD Screen Example 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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