Mathematics
Probability Calculations
Probability calculations involve determining the likelihood of a specific event occurring, often expressed as a number between 0 and 1. This is done by analyzing the possible outcomes and their associated probabilities. Key concepts include the use of formulas such as the probability of an event A given event B, and the addition and multiplication rules for combining probabilities.
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11 Key excerpts on "Probability Calculations"
eBook - PDFStochastic Modeling and Mathematical Statistics
A Text for Statisticians and Quantitative Scientists
- Francisco J. Samaniego(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
1 The Calculus of Probability 1.1 A Bit of Background Most scientific theories evolve from attempts to unify and explain a large collection of in-dividual problems or observations. So it is with the theory of probability, which has been developing at a lively pace over the past 350 years. In virtually every field of inquiry, re-searchers have encountered the need to understand the nature of variability and to quantify their uncertainty about the processes they study. From fields as varied as astronomy, biol-ogy, and economics, individual questions were asked and answered regarding the chances of observing particular kinds of experimental outcomes. Following the axiomatic treatment of probability provided by A. N. Kolmogorov in the 1930s, the theory has blossomed into a separate branch of mathematics, and it is still under vigorous development today. Many of the early problems in mathematical probability dealt with the determination of odds in various games of chance. A rather famous example of this sort is the prob-lem posed by the French nobleman, Antoine Gombaud, Chevalier de M´ er´ e, to one of the outstanding mathematicians of his day, Blaise Pascal: which is the more likely outcome, obtaining at least one six in four rolls of a single die or obtaining at least one double six in twenty-four rolls of a pair of dice? The question itself, dating back to the mid-seventeenth century, seems unimportant today, but it mattered greatly to Gombaud, whose successful wagering depended on having a reliable answer. Through the analysis of questions such as this (which, at this point in time, aptly constitutes a rather straightforward problem at the end of this chapter), the tools of probability computation were discovered. The body of knowledge we refer to as the probability calculus covers the general rules and methods we employ in calculating probabilities of interest. I am going to assume that you have no previous knowledge of probability and statistics.
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 6 Probability Within all of the fields of human intellectual study, mathematics is peculiar in that it deals with certainty by making concrete and precise statements given some equally concrete and precise initial premises. No other subject involves this luxury of the definite. For example, science is fundamentally based on observation of the physical world. Theories are then created to explain the collected data. However, no scientist could honestly claim that a given theory constitutes an absolute truth which would never require ad-justment, or even collapse entirely, under closer scrutiny. History involves the collection of facts concerning past events, but the interpretation of these is always subject to debate. Finally, even though religions claim to deal out absolute truths, it is not an easy matter to convince a sceptic of these certainties! The beauty of mathematics is that any sufficiently expe-rienced, intelligent reader will agree on the truth or falsehood of a proof of a mathematical statement: there can be no question that there are two real solutions to the equation x 2 − 4 = 0, and we certainly do not need to lose any sleep over the result obtained when two is added to three; the answer is five. How then are we to use mathematics to describe events which involve a random factor? In the real world, certainly on the human scale of things, chance events play a major part in life. Probability is the mathematical study of these phenomena. Let us try to understand how we can use the certainties of mathematics to describe random processes. 6.1 The Basic Ideas of Probability We encounter many situations on a daily basis in which an outcome is unknown. Will it rain later today? Will there be a traffic jam on the motorway? Will I have to queue at the bank? Will I win the lottery? On 371 372 A Mathematical Bridge the face of it, some of these questions may seem to be rather difficult even to begin to answer.
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
129 129 CHAPTER 4 We often make statements about probability. For example, a weather forecaster may predict that there is an 80% chance of rain tomorrow. A study may predict that a female, compared to a male, has a higher probability of being in favor of gun control. A college student may ask an instructor about the chances of passing a course or getting an A if he or she did not do well on the midterm examination. Probability, which measures the likelihood that an event will occur, is an important part of statis- tics. It is the basis of inferential statistics, which will be introduced in later chapters. In inferential statistics, we make decisions under conditions of uncertainty. Probability theory is used to evaluate the uncertainty involved in those decisions. For example, estimating next year’s sales for a company is based on many assumptions, some of which may happen to be true and others may not. Probabil- ity theory will help us make decisions under such conditions of imperfect information and uncertainty. Combining probability and probability distributions (which are discussed in Chapters 5 through 7) with descriptive statistics will help us make decisions about populations based on information obtained from samples. This chapter presents the basic concepts of probability and the rules for computing probability. Do you worry about your weight? According to a Gallup poll, 45% of American adults worry all or some of the time about their weight. The poll showed that more women than men worry about their weight all or some of the time. In this poll, 55% of adult women and 35% of adult men said that they worry all or some of the time about their weight.
eBook - PDF- Henry E. Kyburg, Jr, Choh Man Teng(Authors)
- 2001(Publication Date)
- Cambridge University Press(Publisher)
Like the developers of other parts of mathematics at the time, the early probabilists devel- oped the probability calculus in response to real world problems. The mathematics therefore required a real world interpretation—something to tell the user what could be put into the calculus, and to help the user to interpret the results provided by the calculus. There are currently several interpretations of the probability calculus, and we will discuss some of them in Chapter 4. But the original interpretation of the probability calculus is what is now called the classical interpretation. The basic formulation of the classical interpretation (usually attributed to Laplace) is: The probability of an outcome is the ratio of favorable cases to the total number of equally possible cases. The rule is straightforward in some applications. There are six ways in which a die can land; these “ways” are equally possible; dice are manufactured to make this true; the probability of an odd number showing uppermost is therefore the ratio of the number of favorable cases (3) to the total number of cases (6), or 1 2 . But in other cases, the definition is somewhat problematic. Whereas it states what the probability of an outcome is, given the equipossible cases, it does not tell us what the equipossible cases are. The classical probabilists were aware of this problem and attempted to address it by positing the following principle, which we will call the principle of indifference: The elements of a set of outcomes are equally possible if we have no reason to prefer one of them to any other. As we shall see, there are overwhelming difficulties with this principle if we take it generally and attempt to apply it broadly. Nevertheless, in the context of games of chance and other examples in which the basic cases are clearly defined, and are intuitively equipossible or equiprobable, the principle can have heuristic value.
eBook - PDFEvidence in Science
A Simple Account of the Principles of Science for Students of Medicine and Biology
- Kenneth Stone(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
93 CHAPTER XVII MATHEMATICAL PROBABILITY IN RESEARCH MODERN science owes a not inconsiderable debt to the gaming habits of the seventeenth-century French nobility. The beginning of the mathematical theory of probability was with a question put by one of these inveterate gamblers to Pascal, who sent it on to, and discussed it with, Fermât, one of the most brilliant of a brilliant band of seventeenth-century mathematicians. It was an old puzzle. Two equally skilful gamblers have put up a stake, to go to the one who first wins so many points : when the first has scored p points and the second q, they have to break off. How should the stakes be divided? The solution aroused enormous interest, both at the gaming tables and among con-temporary mathematicians. The gamblers wondered if the mathematicians could give them a rule by which they could always win : the mathematicians, painfully aware that they might be useful to a set of useless people, were fascinated by the abstract notion of probability. And so, from dice and cards was conceived the calculus of probability, which Pascal and Fermât are credited with founding. Everyone agrees that mathematical probability is an important concept in modern science. Its usefulness is not impaired by the unexpected fact that no one knows what it means. The distinction between 'credibility' and 'probability' is clear and well understood—the one concerns degrees of belief or doubt, the other a fact about a certain class of events. It is the second, mathematical probability, whose interpretation is in doubt. No mathematician disputes the collection of formulae, described as the calculus of probability, which can be deduced from some simple axioms. Bertrand Russell has pointed out that it often happens that we know a mathematical formula to be true, but can find no meaning, or a number of different meanings, for some of the symbols.
eBook - PDF- Prem S. Mann(Author)
- 2017(Publication Date)
- Wiley(Publisher)
We often make statements about probability. For example, a weather forecaster may predict that there is an 80% chance of rain tomorrow. A study may predict that a female, compared to a male, has a higher probability of being in favor of gun control. A college student may ask an instructor about the chances of passing a course or getting an A if he or she did not do well on the midterm examination. Probability, which measures the likelihood that an event will occur, is an important part of statistics. It is the basis of inferential statistics, which will be introduced in later chapters. In infer- ential statistics, we make decisions under conditions of uncertainty. Probability theory is used to evaluate the uncertainty involved in those decisions. For example, estimating next year’s sales for a company is based on many assumptions, some of which may happen to be true and others may not. Probability theory will help us make decisions under such conditions of imperfect infor- mation and uncertainty. Combining probability and probability distributions (which are discussed in Chapters 5 through 7) with descriptive statistics will help us make decisions about populations based on information obtained from samples. This chapter presents the basic concepts of prob- ability and the rules for computing probability. Do you worry about your weight? According to a Gallup poll, 45% of American adults worry all or some of the time about their weight. The poll showed that more women than men worry about their weight all or some of the time. In this poll, 55% of adult women and 35% of adult men said that they worry all or some of the time about their weight.
eBook - ePub- A. Wolf(Author)
- 2019(Publication Date)
- Routledge(Publisher)
r = 0, according to the nature of the case. In this fuller form we see the basis of the probability of the recurrence of partial correlations or associations, as ascertained by the statistical method of exact enumeration. What has just been said about the probability of inductions by simple enumeration applies here likewise.§ 7. The Use of Calculations of Probability.Lastly, of what practical value is the whole calculus of probability? Some people have very exaggerated notions of its practical value. This is partly due to our respect for figures, in consequence of the important place which mathematics holds in modern science. One tacitly assumes that exact figures express precise knowledge, and one has no suspicion of the ill-defined nebulosities that may masquerade in precise ratios and fractions. Partly, however, the exaggeration is also due to the frequency-theory of probability. This theory rather encourages the confusion of probabilities with frequencies. There is, of course, a connection between them. We have seen that probability can, in many cases, only be calculated on the basis of observed frequencies. Nevertheless, probability and frequency are not identical. Now frequencies, when treated with the necessary precautions, may be of the greatest practical value—as is evident from their use in insurance business and in kindred enterprises. But the calculus of probability is another matter. Probability is concerned even with single events and small groups or series of events, while frequencies always refer to large classes, or long series of events, to what happens “in the long run.” The practical difference is obvious, even in roulette. The bank, doing business with large numbers of players, can rely on frequencies which secure it certain advantages. The individual player, limited to a much smaller number of ventures, simply gambles, and usually pays dearly for his estimates of probability, even when these are based on rational calculations, and are not merely the result of inspiration or superstition. The calculus of probability is, of course, secure of its reputation, just like the ambiguous oracles of antiquity. Whatever happens, the calculus is right. Whether you win or lose, whether you have a long run of good luck, or of ill luck, the calculus is equally right. This may console those who cherish their delusions more than their treasure; but sensible people will not put their trust in such ambiguous oracles. There are, it is true, ingenious gambling systems, based on frequencies rather than on probabilities. But even these systems have their day and cease to be. The best of them is but a snare, cheating the fowler of a bird in the hand for two in the bush. Its validity always depends on “the long run,” which easily outruns the resources of any ordinary individual. For similar reasons, even in legitimate insurance business, the company, relying on frequencies, is on a much better footing than the individual client. But the practical exigencies of life often make it advisable for the individual to take high risks for small amounts rather than incur low risks for large amounts—not to mention the benefits which accrue to the community as a whole from insurance organizations. No precision of figures can eliminate the essential uncertainty of the probable. And, in the last resort, the best method of estimating the probability of anything is by a close examination of the actual conditions. Even insurance companies do not rely entirely on frequencies, but have each case examined by an expert (doctor, or engineer, etc.), according to the nature of the risk. This is done primarily in order to determine accurately to what precise class the risk belongs, since each class has its own frequency. But that is not the whole explanation. For, if everything were taken into account indiscriminately, each case would be sui generis
eBook - PDFPrimary Mathematics
Integrating Theory with Practice
- Penelope Serow, Rosemary Callingham, Tracey Muir(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
1. Identify the language of probability used on these websites. 2. What is the difference between expressing probability as the chance of an event and as the odds of something happening? 3. How is probability expressed mathematically? (You will see uncertainty expressed as odds and chance presented as a percentage, a fraction, a ratio or in words.) 4. Think about what makes probability tricky for children (as well as many adults). Discuss your ideas with another person. These sites indicate the pervasiveness of uncertainty and the complexity associated with expressing it. It is therefore not surprising that this is one of the trickiest areas of the mathematics curriculum. It is, however, also one of the most important, because so many everyday situations are expressed in probabilistic terms, such as the risk of dying from a particular disease, insurance and investment risks, or the odds of a par- ticular team winning in a sporting contest. Investigative approaches to developing an understanding of probabil- ity are useful for children. Using approaches where children collect data about random events, such as coin tossing or rolling a dice, helps to develop the ideas of uncertain outcomes using a relative frequency approach. Such an approach is relevant for all children because it allows them to build their knowledge on the basis of experience, and helps them to develop new, sound intuitions necessary for better understanding probability. Outcomes the set of possible predictable events in a given situation, such as heads and tails when tossing a single coin Relative frequency the number of times a given outcome occurs relative to other possible outcomes; this is the basis for classical probability theory Weblinks 9.1–9.3 190 PRIMARY MATHEMATICS Understanding probability Bryant and Nunes (2012) suggest that children need to understand four key concepts in probability: randomness, sample space, quantifying probability and correlations.
eBook - PDFBayesian Data Analysis for the Behavioral and Neural Sciences
Non-Calculus Fundamentals
- Todd E. Hudson(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
CHAPTER TWO MECHANICS OF Probability Calculations The goal of this chapter is to provide the practical machinery needed for making Probability Calculations and assigning probability distributions. This involves both alge- braically manipulating probability statements and manipulating the logical expressions within those probability statements. We therefore begin with a discussion of symbolic logic, which will provide us the necessary skill in interpreting and manipulating the log- ical expressions within probability statements. By the end of the chapter we will use this machinery to assign probability distributions, and investigate the role of sampling dis- tributions and likelihood functions – two types of probability function – in scientific inference. Along the way (Section 2.1), we discuss the rationale for the conditioning statements found in the product rule that can sometimes seem opaque, and introduce the concepts of marginalization and extending the question in an applied context. Section 2.2 provides an introduction to assigning probability distributions, focusing on sampling distributions that describe sequences of coin tosses and other two-outcome processes (called Bernoulli trials). Section 2.3 describes the relationship between the sampling dis- tribution, which is the crucial element in the null-hypothesis testing algorithm, and the likelihood function, used in Bayesian statistics. This section also provides a compari- son of the potential for error when inferences are based on sampling distributions as compared to when they are based on likelihood functions. As with any area of mathematics, your understanding of probability will improve with practice.
eBook - PDFStatistical Methods in the Atmospheric Sciences
An Introduction
- Daniel S. Wilks(Author)
- 1995(Publication Date)
- Academic Press(Publisher)
Chapter 2 Review of Probability 2.1 Background The material in this chapter is meant to be a brief review of the basic elements of probability. More complete treatments of the basics of probability can be found in any good introductory statistics text, such as Dixon and Massey's (1983) Intro- duction to Statistical Analysis or Winkler's (1972) Introduction to Bayesian Infer- ence and Decision. Our uncertainty about the atmosphere, or about any other system, for that mat- ter, is of different degrees in different instances. For example, you cannot be com- pletely certain whether rain will occur at your home tomorrow, or whether the average temperature next month will be greater or less than the average tempera- ture last month. But it is likely that you are more sure about the latter question than about the former one. It is not sufficient, or even particularly informative, to say that a particular event is uncertain. Rather, one is faced with the problem of expressing or characteriz- ing degrees of uncertainty. A possible approach is to use qualitative descriptors such as likely, unlikely, .... possible, or chance of. Conveying uncertainty through such phrases, however, is ambiguous and open to varying interpretations (Murphy and Brown, 1983). For example, it is not clear whether rain likely or rain probable indicates less uncertainty about the prospects for rain. It is generally preferable to express uncertainty quantitatively, and this is done using numbers called probabilities. In a limited sense, probability is no more than an abstract mathematical system that can be developed logically from three prem- ises called the axioms of probability. This system would b.e uninteresting to many people, including perhaps yourself, except that the resulting abstract concepts are relevant to real-world systems involving uncertainty. Before presenting the axi- oms of probability and a few of their more important implications, it is necessary to define some terminology.
eBook - PDFReliability Engineering and Risk Analysis
A Practical Guide, Third Edition
- Mohammad Modarres, Mark P. Kaminskiy, Vasiliy Krivtsov(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
15 2 Basic Reliability Mathematics Review of Probability and Statistics 2.1 INTRODUCTION In this chapter, we discuss the elements of mathematical theory that are relevant to the study of the reliability of physical objects. We begin with a presentation of basic concepts of probability. Then, we briefly consider some fundamental concepts of statistics that are used in reliability analysis. 2.2 ELEMENTS OF PROBABILITY Probability is a concept that people use formally and casually every day. Weather forecasts are probabilistic in nature. People use probability in their casual conversations to show their perception of the likely occurrence or nonoccurrence of particular events. Odds are given for the outcomes of sport events and are used in gambling. The formal use of probability concepts is widespread in sci-ence, in astronomy, biology, and engineering. In this chapter, we discuss the formal application of probability theory in the field of reliability engineering. 2.2.1 S ETS AND B OOLEAN A LGEBRA To perform operations associated with probability, it is often necessary to use sets. A set is a col-lection of items or elements, each with some specific characteristics. A set that includes all items of interest is referred to as a universal set , denoted by Ω . A subset refers to a collection of items that belong to a universal set. For example, if set Ω represents the collection of all pumps in a power plant, then the collection of electrically-driven pumps is a subset E of Ω . Graphically, the relation-ship between subsets and sets can be illustrated through Venn diagrams. The Venn diagram in Figure 2.1 shows the universal set Ω by a rectangle and subsets E 1 and E 2 by circles. It can also be seen that E 2 is a subset of E 1 . The relationship between subsets E 1 and E 2 and the universal set can be symbolized by E 2 ⊂ E 1 ⊂ Ω .
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