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Quality Improvement Chapter 8- Fundamentals of Probability PowerPoint presentation to accompany Besterfield, Quality Improvement, 9e Outline Definition of Probability Theorems of Probability Counting of Events Discrete Probability Distributions Continuous Probability Distribution Quality Improvement, 9e Dale H. Besterfield 2 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Learning Objectives 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. Quality Improvement, 9e Dale H. Besterfield 3 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Learning Objectives cont’d. 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. Quality Improvement, 9e Dale H. Besterfield 4 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability 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. Quality Improvement, 9e Dale H. Besterfield 5 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability 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. Quality Improvement, 9e Dale H. Besterfield 6 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability 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). Quality Improvement, 9e Dale H. Besterfield 7 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability When the number of outcomes is known or when the number of outcomes is found by experimentation: P(A) = NA/N where: P(A) = probability of event A ocurring to 3 decimal places NA = number of successful outcomes of event A N = total number of possible outcomes Quality Improvement, 9e Dale H. Besterfield 8 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability 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. Quality Improvement, 9e Dale H. Besterfield 9 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Definition of Probability In the infinite situation the probability of an event occurring is proportional to the population distribution. Quality Improvement, 9e Dale H. Besterfield 10 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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. Quality Improvement, 9e Dale H. Besterfield 11 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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) Quality Improvement, 9e Dale H. Besterfield 12 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability One Event Out or Two or More Events Mutually Exclusive Not Mutually Exclusive Theorem 3 Theorem 4 Two or More Event Out or Two or More Events Independent Dependent Theorem 6 Theorem 7 Figure 7-2 When to use Theorems 3,4,6 and 7 Quality Improvement, 9e Dale H. Besterfield 13 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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”. Quality Improvement, 9e Dale H. Besterfield 14 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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 Quality Improvement, 9e Dale H. Besterfield 15 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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”. Quality Improvement, 9e Dale H. Besterfield 18 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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 Quality Improvement, 9e Dale H. Besterfield 19 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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 Quality Improvement, 9e Dale H. Besterfield 20 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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) Quality Improvement, 9e Dale H. Besterfield 21 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Theorems of Probability 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. Quality Improvement, 9e Dale H. Besterfield 22 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Counting of Events 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. Quality Improvement, 9e Dale H. Besterfield 23 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Counting of Events 2. Permutations A permutation is an ordered arrangement of a set of objects. n! P (n r )! n r Example: The word “cup”…… cup, cpu, upc, ucp, puc, and pcu. Quality Improvement, 9e Dale H. Besterfield 24 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Counting of Events 3. Combinations If the way the objects are ordered is unimportant, then we have a combination: n! C r !(n r )! n r 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. Quality Improvement, 9e Dale H. Besterfield 25 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. n(n 1) n2 2 n ( p q) p np q p q ......... q 2 n Quality Improvement, 9e Dale H. Besterfield n n 1 27 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions Figure 8-6 Distribution of the number of tails for an infinite number of tosses of 11 coins Quality Improvement, 9e Dale H. Besterfield 28 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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 n! d n d P(d ) p0 q0 d !(n d )! Quality Improvement, 9e Dale H. Besterfield 29 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. Quality Improvement, 9e Dale H. Besterfield 30 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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 Quality Improvement, 9e Dale H. Besterfield 31 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. Quality Improvement, 9e Dale H. Besterfield 32 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. c (np0 ) np0 P( c ) e c! Quality Improvement, 9e Dale H. Besterfield 33 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. Quality Improvement, 9e Dale H. Besterfield 34 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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). Quality Improvement, 9e Dale H. Besterfield 35 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Discrete Probability Distributions 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. Quality Improvement, 9e Dale H. Besterfield 36 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved Continuous Probability Distributions 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. Quality Improvement, 9e Dale H. Besterfield 37 © 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 Quality Improvement, 9e Dale H. Besterfield 39 © 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 Quality Improvement, 9e Dale H. Besterfield 40 © 2013, 2008 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 • All Rights Reserved