1. Quality Improvement PowerPoint presentation to accompany Besterfield, Quality Improvement, 9e PowerPoint presentation to accompany Besterfield, Quality Improvement, 9e Chapter 8- Fundamentals of Probability

2. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 2 Outline   Definition of Probability   Theorems of Probability   Counting of Events   Discrete Probability Distributions   Continuous Probability Distribution

3. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 3 When you have completed this chapter you should be able to:  Define probability using the frequency definition.  Know the seven basic theorems of probability.  Identify the various discrete and continuous probability distributions. Learning Objectives

4. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 4 When you have completed this chapter you should be able to:  Calculate the probability of non-conforming units occuring using the Hypergeometric, Binomial and Poisson distributions.  Know when to use the Hypergeometric, Binomial and Poisson distributions. Learning Objectives cont’d.

5. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 5  Likelihood, chance, tendency, and trend.  The chance that something will happen.  If a Nickel is tossed, the probability of a head is 1/2 and the probability of the tail is 1/2.  When a die is tossed, the probability of one spot is 1/6, the probability of two spots is 1/6,.....  Drawing a card from a deck of cards. The probability of a spade is 13/52. Definition of Probability

6. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 6  The area of each distribution is equal to 1.  The area under the normal distribution curve, which is a probability distribution, is equal to 1.  The total probability of any situation will be equal to 1. Definition of Probability

7. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 7  The probability is expressed as a decimal (the probability of a head is 0.5).  An event is a collection of outcomes (six-sided die has six possible outcomes). Definition of Probability

8. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 8 When the number of outcomes is known or when the number of outcomes is found by experimentation: P(A) = N A /N where: P(A) = probability of event A ocurring to 3 decimal places N A = number of successful outcomes of event A N = total number of possible outcomes Definition of Probability

9. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 9  The probability calculated using known outcomes is the true probability, and the one calculated using experimental outcomes is different due to the chance factor.  For an infinite situation (N = ∞), the definition would always lead to a probability of zero. Definition of Probability

10. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 10. In the infinite situation the probability of an event occurring is proportional to the population distribution. Definition of Probability

11. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 11 Theorem 1 Probability is expressed as a number between 1 and 0, where a value of 1 is a certainty that an event will occur and a value of 0 is a certainty that an event will not occur. Theorems of Probability

12. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 12 Theorem 2 If P(A) is the probability that event A will occur, then the probability that A will not occur is: P(notA) = 1- P(A) Theorems of Probability

13. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 13 One Event Out or Two or More Events Mutually Exclusive Theorem 3 Not Mutually Exclusive Theorem 4 Two or More Event Out or Two or More Events Independent Theorem 6 Dependent Theorem 7 Theorems of Probability Figure 7-2 When to use Theorems 3,4,6 and 7

14. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 14 Theorem 3 If A and B are two mutually exclusive events (the occurrence of one event makes the other event impossible), then the probability that either event A or event B will occur is the sum of their respective probabilities: P(A or B) = P(A) +P(B) This is the “additive law of probability”. Theorems of Probability

15. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 15 Theorem 4 If event A and event B are not mutually exclusive, then the probability of either event A or event B or both is given by: P(A or B or both) = P(A) +P(B) – P(both) Events that are not mutually exclusive have some outcomes in common Theorems of Probability

16. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 16 Theorem 2 If P(A) is the probability that event A will occur, then the probability that A will not occur is: P(notA) = 1- P(A) Theorems of Probability

17. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 17 One Event Out or Two or More Events Mutually Exclusive Theorem 3 Not Mutually Exclusive Theorem 4 Two or More Event Out or Two or More Events Independent Theorem 6 Dependent Theorem 7 Theorems of Probability Figure 8-2 When to use Theorems 3,4,6 and 7

18. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 18 Theorem 3 If A and B are two mutually exclusive events (the occurrence of one event makes the other event impossible), then the probability that either event A or event B will occur is the sum of their respective probabilities: P(A or B) = P(A) +P(B) This is the “additive law of probability”. Theorems of Probability

19. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 19 Theorem 4 If event A and event B are not mutually exclusive, then the probability of either event A or event B or both is given by: P(A or B or both) = P(A) +P(B) – P(both) Events that are not mutually exclusive have some outcomes in common Theorems of Probability

20. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 20 Theorem 5 The sum of the probabilities of the events of a situation is equal to 1.000 P(A) + P(B) + …..+ P(N) = 1.000 Theorems of Probability

21. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 21 Theorem 6 If A and B are independent events (one where its occurrence has no influence on the probability of the other event or events), then the probability of both A and B occurring is the product of their respective probabilities: P(A and B) = P(A) X P(B) Theorems of Probability

22. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 22 Theorem 7 If A and B are dependent events, the probability of both A and B occurring is the probability of A and the probability that if A occurred, then B will occur also: P(A and B) = P(A) X P(B\A) P(B\A) is defined as the probability of event B, provided that event A has ocurred. Theorems of Probability

23. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 23 1. Simple multiplication If an event A can happen in any of a ways or outcomes and, after it has occurred, another event B can happened in b ways or outcomes, the number of ways that both events can happen is ab. Counting of Events

24. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 24 2. Permutations A permutation is an ordered arrangement of a set of objects. Example: The word “cup”…… cup, cpu, upc, ucp, puc, and pcu. Counting of Events

25. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 25 3. Combinations If the way the objects are ordered is unimportant, then we have a combination: Counting of Events Example: The word “cup” has 6 permutations when the 3 objects are taken 3 at a time. There is only one combination, since the same three letters are in different order.

26. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 26 Hypergeometric Probability Distribution 1. Occurs when the population is finite and the random sample is taken without replacement. 2. The formula is constructed of 3 combinations (total, nonconforming, and conforming): Discrete Probability Distributions

27. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 27 Binomial Probability Distribution 1. It is applicable to discrete probability problems that have an infinite number of items or that have a steady stream of items coming from a work center.. 2. It is applied to problems that have attributes. Discrete Probability Distributions

28. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 28 Discrete Probability Distributions Figure 8-6 Distribution of the number of tails for an infinite number of tosses of 11 coins

29. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 29 Binomial Probability Distribution cont’d 3. See Figure 8-6. Since p=q, the distribution is symmetrical regardless of the value of n, however, when p is not equal to q, the distribution is asymmetrical. 4. In quality work p is the portion or fraction nonconforming and is usually less than 0.15 Discrete Probability Distribution s

30. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 30 Binomial Probability Distribution cont’d. 5. As the sample size gets larger, the shape of the curve will become symmetrical even though p is not equal to q. 6. It requires that there be two and only two possible outcomes (C, NC) and that the probability of each outcome does not change. Discrete Probability Distributions

31. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 31 Binomial Probability Distribution cont’d. 7. The use of the binomial requires that the trials be independent. 8. It can be approximated by the Poisson when Po≤0.10 and nPo≤5. 9. The normal curve is an excellent approximation when Po is close to 0.5 and n>̳ 10 or Discrete Probability Distributions

32. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 32 Poisson Probability Distribution 1. It is applicable to many situations that involve observations per unit of time. 2. It is also applicable to situations involving observations per unit amount. 3. In each of the preceding situations, there are many equal opportunities for the occurrence of an event. Discrete Probability Distributions

33. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 33 Poisson Probability Distribution cont’d. 4. The Poisson is applicable when n is quite large and Po is small.. 5. When Poisson is used as an approximation to the binomial, the symbol c has the same meaning as d has in the binomial and hypergeometric formulas. Discrete Probability Distributions

34. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 34 Poisson Probability Distribution cont’d. 6. When nPo gets larger, the distribution approaches symmetry. 7. Table C in the Appendix. 8. The Poisson probability is the basis for attribute control charts and for acceptance sampling. Discrete Probability Distributions

35. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 35 Poisson Probability Distribution cont’d. 9. It is used in other industrial situations, such as accident frequencies, computer simulation, operations research, and work sampling. 10. Uniform (generate a random number table), Geometric, and Negative binomial (reliability studies for discrete data). Discrete Probability Distributions

36. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 36 Poisson Probability Distribution cont’d. 11. The Poisson can be easily calculated using Table C. 12. Similarity among the hypergeometric, binomial, and Poisson distributions can exist. Discrete Probability Distributions

37. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 37 Normal Probability Distribution 1. When we have measurable data. 2. The normal curve is a continuous probability distribution.. 3. Under certain condition the normal probability distribution will approximate the binomial probability distribution. Continuous Probability Distributions

38. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved 38 Normal Probability Distribution cont’d. 4. The Exponential probability distribution is used in reliability studies when there is a constant failure rate. 5. The Weibull distribution is used when the time to failure is not constant. Continuous Probability Distributions

39. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Computer Program  Microsoft EXCEL/Minitab will solve for permutations, combinations, hypergeometric, binomial, and Poisson 39

40. Quality Improvement, 9e Dale H. Besterfield © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Homework  33, 35, 37, 41, 43 by hand and minitab 40