Technology & Engineering
Addition Rule of Probability
The Addition Rule of Probability states that the probability of the occurrence of either one of two mutually exclusive events is the sum of their individual probabilities. In other words, if two events cannot occur at the same time, the probability of one or the other occurring is the sum of their individual probabilities.
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7 Key excerpts on "Addition Rule of Probability"
No longer available |Learn moreUnderstandable Statistics
Concepts and Methods, Enhanced
- Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Other important ways will be discussed later. • The law of large numbers indicates that as the number of trials of a statistical experiment or observation increases, the relative frequency of a designated event becomes closer to the theoretical probability of that event. • Events are mutually exclusive if they cannot occur together. Events are independent if the occurrence of one event does not change the probability of the occurrence of the other. • Conditional probability is the probability that one event will occur, given that another event has occurred. • The complement rule gives the probability that an event will not occur. The addition rule gives the probability that at least one of two specified events will occur. The multiplication rule gives the probability that two events will occur together. • To determine the probability of equally likely events, we need to know how many outcomes are possible. Devices such as tree diagrams and counting rules—such as the multiplication rule of counting, the permutations rule, and the combinations rule—help us determine the total number of outcomes of a statistical experiment or observation. In most of the statistical applications of later chapters, we will use the addition rule for mutu-ally exclusive events and the multiplication rule for independent events. Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDFBusiness Statistics
For Contemporary Decision Making
- Ken Black(Author)
- 2020(Publication Date)
- Wiley(Publisher)
4.4 Addition Laws 103 apply more readily in certain situations than in others. No best method is available for solving all probability problems. In some instances, the joint probability table lays out a problem in a readily solvable manner. In other cases, setting up the joint probability table is more difficult than solving the problem in another way. The probability laws almost always can be used to solve probability problems. Four laws of probability are presented in this chapter: the addition laws, the multiplica-tion laws, conditional probability, and Bayes’ rule. The addition laws and the multiplication laws each have a general law and a special law. 4.4 Addition Laws The general law of addition is used to find the probability of the union of two events, P ( X ∪ Y ). The expression P ( X ∪ Y ) denotes the probability of X occurring or Y occurring or both X and Y occurring. General Law of Addition P ( X ∪ Y ) = P ( X ) + P ( Y ) − P ( X ∩ Y ) where X , Y are events and ( X ∩ Y ) is the intersection of X and Y . Yankelovich Partners conducted a survey for the American Society of Interior Designers in which workers were asked which changes in office design would increase productivity. Re-spondents were allowed to answer more than one type of design change. The number one change that 70% of the workers said would increase productivity was reducing noise. In second place was more storage/filing space, selected by 67%. If one of the survey respondents was ran-domly selected and asked what office design changes would increase worker productivity, what is the probability that this person would select reducing noise or more storage/filing space? Let N represent the event “reducing noise.” Let S represent the event “more storage/ filing space.” The probability of a person responding with N or S can be symbolized statistical-ly as a union probability by using the law of addition.
eBook - ePub- Dennis V. Lindley(Author)
- 2013(Publication Date)
- Wiley(Publisher)
With this terminology, the last form of the multiplication rule reads that your probability of the product of two independent events is the product of their separate probabilities, a result that is attractive because it is easy to remember. Unfortunately, it is true only if the events are independent; otherwise it is wrong, and often seriously wrong. Similarly, (5.2) reads that your probability of the sum of two events is the sum of their separate probabilities. Again, this is true only under restrictions, but this time the restriction is not independence but the requirement that the events be exclusive. Simple as these special forms are, their simplicity can easily lead to errors and are therefore best avoided unless the restrictions that made them valid are always remembered throughout the calculations. The desire for simplicity has often been emphasized, but here is an example where it is possible to go too far and think of the addition and multiplication rules in their simpler forms, forgetting the restrictions that must hold before they are correct. Notice that the restriction, necessary for the simple form of the addition rule, that the events be exclusive, or the disjunction impossible, is a logical restriction, having nothing to do with uncertainty, whereas independence, the restriction with the multiplication rule, is essentially probabilistic. It is perhaps pedantic to point out that the simple form of the addition rule is correct if you judge the disjunction to have probability zero, rather than knowing it is logically impossible, but we will see in §6.8 that it is dangerous to attach probability zero to anything other than a logical impossibility. 5.4 The Basic Rules There are now two rules that your probabilities have to obey: addition and multiplication
eBook - PDFAnalytical Methods for Risk Management
A Systems Engineering Perspective
- Paul R. Garvey(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
Chapter 2 Elements of Probability Theory 2.1 Introduction Whether referring to a storm’s intensity, an arrival time, or the success of a decision, the word “probable,” or “likely,” has long been part of our language. Most people have an appreciation for the impact of chance on the occurrence of an event. In the last 350 years, the theory of probability has evolved to explain the nature of chance and how it can be studied. Probability theory is the formal study of events whose outcomes are uncertain. Its origins trace to 17th-century gambling problems. Games that involved playing cards, roulette wheels, and dice provided mathematicians with a host of interest-ing problems. The solutions to many of these problems yielded the first principles of modern probability theory. Today, probability theory is of fundamental impor-tance in science, engineering, and business. Engineering risk management aims to identify and manage events whose out-comes are uncertain. In particular, its focus is on events that, if they occur, have unwanted impacts or consequences to a project or program. The phrase “ if they occur ” means these events are probabilistic in nature. Thus, understanding them in the context of probability concepts is essential. This chapter presents an in-troduction to these concepts and illustrates how they apply to managing risks in engineering systems. 2.2 Interpretations and Axioms We begin this discussion with the traditional look at dice. If a six-sided die is tossed, there clearly are six possible outcomes for the number that appears on the upturned face. These outcomes can be listed as elements in a set { 1 , 2 , 3 , 4 , 5 , 6 } . 13 14 Elements of Probability Theory The set of all possible outcomes of an experiment, such as tossing a six-sided die, is called the sample space , which we will denote by . The individual outcomes of are called sample points, which we will denote by ω .
eBook - PDFBusiness Statistics
For Contemporary Decision Making
- Ken Black(Author)
- 2023(Publication Date)
- Wiley(Publisher)
These tools include sample space, tree diagrams, Venn diagrams, the laws of probability, joint probability tables, and insight. Because of the individuality and variety of probability problems, some techniques apply more readily in certain situations than in others. No best method is available for solving all probability problems. In some instances, the joint probability table lays out a problem in a readily solvable manner. In other cases, setting up the joint probability table is more difficult than solving the problem in another way. The probability laws almost always can be used to solve probability problems. Four laws of probability are presented in this chapter: the addition laws, the multiplica- tion laws, conditional probability, and Bayes’ rule. The addition laws and the multiplication laws each have a general law and a special law. 4.4 Addition Laws LEARNING OBJECTIVE Calculate probabilities using the general law of addition, along with a joint probability table, the complement of a union, or the special law of addition if necessary. Lecture Video The general law of addition is used to find the probability of the union of two events, P(X ∪ Y). The expression P(X ∪ Y) denotes the probability of X occurring or Y occurring or both X and Y occurring. General Law of Addition P(X ∪ Y ) = P(X ) + P(Y ) − P(X ∩ Y ) where X, Y are events and (X ∩ Y ) is the intersection of X and Y. Yankelovich Partners conducted a survey for the American Society of Interior Designers in which workers were asked which changes in office design would increase productivity. Respon- dents were allowed to answer more than one type of design change. The number one change Thinking Critically About Statistics in Business Today Probabilities in the Dry-Cleaning Business According to the International Fabricare Institute, about two- thirds or 67% of all dry-cleaning customers are female, and 65% are married.
eBook - PDFRational Descriptions, Decisions and Designs
Pergamon Unified Engineering Series
- Myron Tribus, Thomas F. Irvine, James P. Hartnett(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
The joint rule (product rule) and the denial rule together furnish enough informa-tion for the computation of the probability of an or statement from the individual probabilities. To begin, define the statement D as: 37 RATIONAL DESCRIPTIONS, DECISIONS AND DESIGNS D = A + B so that d = ab. From the denial rule we have: p ( D | E ) = l -p ( d | E ) or p(A + B|E)= 1 -p(ab|E). From the product rule, p(A + B|E) = 1 - p(a|bE)p(b|E). From the denial rule we may substitute for p(a | bE): p(A + B|E) = 1 - [1 - p(A|bE)]p(b|E) and from the product rule we find p(A + B|E) = 1 - p(b|E) + p(Ab|E). The first two terms on the right hand side may be combined using the denial rule: p(A + B|E) = p(B|E) + p(Ab|E). From the product rule we have p(A + B|E) = p(B|E) + p(b|AE)p(A|E). Using the denial rule on p(b | AE) we find p(A + B|E) = p(B|E) + [l-p(B|AE)]p(A|E). Multiplying through, rearranging terms and using the product rule gives the final result: p(A + B|E) = p(A|E) 4- p(B|E) - p(AB|E). I l l Equation II-1 is called the additive rule for probabilities. It is a simple consequence of eqn. 1-19 and 1-20. A GEOMETRICAL INTERPRETATION OF THE ADDITIVE RULE The following two equations represent different versions of the denial rule: p(A|BE) + p(a|BE)= 1, p(A|bE) + p(ajbE)= 1. 38 SOME MATHEMATICAL PRELIMINARIES Multiply the first equation by p(B|E) and the second by p(b|E). Use the product rule to combine the products on the left side of each equation. Upon adding the result it is found that: p(AB|E) + p(aB|E) + p(Ab|E) + p(ab|E) = 1. This equation has a geometric interpretation. Consider the figure below in which the unit square has been partitioned into areas representative Fig. II-1. Venn diagram for probabilities. of the above terms. Upon transposing the term p(ab|E) to the right in the above equation, and using the denial rule in the form p(A + B|E)= 1 -p(ab|E) the validity of eqn.
- Brase/Brase, Charles Henry Brase, Corrinne Pellillo Brase(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
More than two mutually exclusive events EXAMPLE 7 The addition rule extends to more than two events that are not mutually exclusive, and the multiplication rule extends to more than two events that are not independent as well. However, the formulas for these extensions are more complicated than the formulas given in the text for the restricted conditions of mutually exclusive or independent. Surveys are very popular with students. Surveys are something they can relate to, and the prob-lem solutions are relatively straightforward. Contingency table EXAMPLE 8 LOOKING FORWARD Chapters 6 and 7 involve several probability distributions. In these chapters, many of the events of interest are mutually exclusive or in-dependent. This means we can compute prob-abilities by using the extended addition rule for mutually exclusive events and the extended multiplication rule for independent events. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 212 Chapter 5 ELEMENTARY PROBABILITY THEORY Suppose an employee is selected at random from the 140 Hopewell employ-ees. Let us use the following notation to represent different events of choosing: E executive; PW production worker; D Democrat; R Republican; I Independent. (a) Compute P ( D ) and P ( E ). SOLUTION: To find these probabilities, we look at the entire sample space. P 1 D 2 5 Number of Democrats Number of employees 5 68 140 < 0.48 6 P 1 E 2 5 Number of executives Number of employees 5 48 140 < 0.34 3 (b) Compute P 1 D | E 2 .
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