Mathematics
Probability
Probability is a measure of the likelihood of a specific event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In mathematics, probability theory is used to analyze random phenomena and make predictions based on the likelihood of different outcomes.
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11 Key excerpts on "Probability"
eBook - PDFQuantitative Techniques in Business, Management and Finance
A Case-Study Approach
- Umeshkumar Dubey, D P Kothari, G K Awari(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
The set S of all possible outcomes (or elementary events) of a given experiment is called the sample space of the experiment. 5.3 Probability The development of the theory of Probability dates back to the seventeenth century. The Probability formulae and techniques were developed by Jacob Bernoulli (1713), De Moivre 107 Probability Theory (1718) and Thomas Bayes (1764). Most of these were concerned with the application of the theory of permutations and combinations to the calculation of probabilities associated with various dice and card games. Probability means ‘a chance’. It can be defined as an expression of chance of occurrence of an event. Probability is used in making viable predictions, suitable decisions, convenient planning and operational policies. Thus, Probability is one of the significant contributors to the science of quantitative techniques. The degree of uncertainty can be measured numerically with the help of Probability. Probability theory is used to analyse data for decision making. 1. The insurance industry uses Probability to calculate premium rates. 2. A stock analyst/investor uses Probability to estimate the returns of the stocks. 3. A project manager uses Probability in decision making. Probability is the chance of occurrence or a number assigned to the occurrence of an event, in a sample space. Probability means ‘a chance’. The Probability of an event equals the number times it happens divided by the number of opportunities. It can be defined as an expression of the chance of occurrence of an event. Business activ-ities are much associated with future changes. The future is the most uncertain element. Therefore, it is still a dream to forecast the future with 100% certainty in any decision prob-lem. The Probability theory provides a media of coping with uncertainty. Probability is used to make viable predictions, suitable decisions, convenient planning and operational policies.
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 - PDFEntropy Demystified: The Second Law Reduced To Plain Common Sense
The Second Law Reduced to Plain Common Sense
- Arieh Ben-naim(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
We shall use the term Probability as used in physics. We shall always discuss events, not propositions. We shall not deal with the question of the meaning of Probability, randomness, etc. These questions encroach upon philosophy. As we shall discuss in this chapter, questions about Probability are always about conditional Probability. Sometimes, the condition is formulated in terms of an event that has occurred or will occur. Other times, the condition can contain whatever information, or knowledge given on the event. Without any knowledge, no answer can be given to any question of Probability. Introduction to Probability Theory, Information Theory, and all the Rest 21 quantitative and constitutes an objective branch 3 of mathemat-ics. Although it is not applicable to all possible events, proba-bility is applicable to a very large body of events; for instance, games of chance and many “events” which are the outcomes of experiments in physics. Thus, if you claim that there is a Probability of 90% that the Messiah will appear on Monday next week, and I claim that the chances are only 1%, there is no way to decide who is right or wrong. In fact, even if we wait for the coming Monday and see that nothing has happened, we could not tell whose estimate of the Probability was correct. 4 However, for some classes of well-defined experiments, there is a Probability that “ belongs ” to the event, and that Probability is accepted by all. For instance, if we toss a coin which we have no reason to suspect to be unbalanced or “unfair”, the odds for the outcomes head ( H ) or tail ( T ) are 50%:50%, respectively. In essence, there is no proof that these are the “correct” probabilities. One can adopt a practical experimental proof based on actual, numerous 3 It should be noted that “objective” here, does not imply “absolute Probability.” Any Probability is a “conditional Probability,” i.e., given some information, or evidence.
eBook - PDF- Bruce Hansen(Author)
- 2022(Publication Date)
- Princeton University Press(Publisher)
CHAPTER 1 BASIC Probability THEORY 1.1 INTRODUCTION Probability theory is foundational for economics and econometrics. Probability is the mathematical lan- guage used to handle uncertainty, which is central for modern economic theory. Probability theory is also the foundation of mathematical statistics, which is the foundation of econometric theory. Probability is used to model uncertainty, variability, and randomness. When we say that something is “uncertain”, we mean that the outcome is unknown. For example, how many students will there be in next year’s Ph.D. entering class at your university? “Variability” means that the outcome is not the same across all occurrences. For example, the number of Ph.D. students fluctuates from year to year. “Randomness” means that the variability has some sort of pattern. For example, the number of Ph.D. students may fluctuate between 20 and 30, with 25 more likely than either 20 or 30. Probability gives us a mathematical language to describe uncertainty, variability, and randomness. 1.2 OUTCOMES AND EVENTS Suppose you take a coin, flip it in the air, and let it land on the ground. What will happen? Will the result be “heads” (H) or “tails” (T)? We do not know the result in advance, so we describe the outcome as random. Suppose you record the change in the value of a stock index over a period of time. Will the value increase or decrease? Again, we do not know the result in advance, so we describe the outcome as random. Suppose you select an individual at random and survey them about their economic situation. What is their hourly wage? We do not know in advance. The lack of foreknowledge leads us to describe the outcome as random. We will use the following terms. An outcome is a specific result. For example, in a coin flip, an outcome is either H or T. If two coins are flipped in sequence, we can write an outcome as HT for a head and then a tails. A roll of a six-sided die has the six outcomes {1, 2, 3, 4, 5, 6}.
eBook - PDFPrimary Mathematics
Integrating Theory with Practice
- Penelope Serow, Rosemary Callingham, Tracey Muir(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Exploring chance and Probability 9 Learning outcomes By the end of this chapter, you will: • understand the difference between objective and subjective views of Probability • be able to use a range of random generators to determine probabilities • recognise the applications of Probability in daily life • be confident in using technology effectively to develop ideas about uncertainty. Introduction Uncertainty is a part of everyday life. We live with a range of situations that inherently have an element of uncertainty in them – for example, crossing the road, going on holiday, or making a major purchase – but we often ignore the embedded chance in these activities. Risk is acknowledged in many activities, and much effort is expended in identifying these risks and minimising any potential negative outcomes. In schools, for example, a risk assessment is required prior to any excursion with children. Probability is the strand of mathematics that addresses uncertainty. This chapter explores ideas relating to Probability in the mathematics classroom. Why is Probability important? Probability is concerned with how likely it is that something will happen or that a particular outcome will be achieved. As such, it can be both objective – as in calculating the Probability of tossing a six on a fair dice – or subjective – as in assigning a Probability to the chance of rain. In both examples, there is some uncertainty attached, and Probability aims to quantify that uncertainty. Both objective and subjective views of Probability are met in the primary mathematics classroom. As a teacher, it is important to understand the ideas of Probability, but for many people these are difficult to grasp. The formal ideas challenge our intuitions, even for experienced and knowledgeable people. There is a famous Probability problem based on a game show called ‘Monty Hall’s problem’.
- Aliakbar Montazer Haghighi, Indika Wickramasinghe(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
51 2 Basics of Probability 2.1 BASICS OF Probability In this book, unless otherwise stated, we will assume that occurrences of events are not deterministic. In other words, the events cannot be entirely determined by the initial states and inputs. Instead, they are perceived to be random, probabilistic, or stochastic. Nature of these occurrences of events is random, probabilistic, or stochas-tic. We will start with basic definitions and develop, in the process, the vocabulary needed to discuss the subject matter with the clarity sufficient even to address the more advanced materials to be encountered later in this book. Definition 2.1 When an experiment is performed, its results are referred to as the outcomes . If such results are not deterministic, the experiment is called a chance experiment , a random experiment , or a trial . Thus, from now to the end of this book, we will be considering random experi-ments with uncertain outcomes. Definition 2.2 A set of outcomes is called an event . An event with only one outcome, that is, a sin-gleton, is referred to as an elementary or simple event . Hence, in general, an event is a collection of simple events. An event may be defined as an element of a σ -field, denoted by 𝕊 ( defined in Chapter 1) of subsets of the sample space Ω . Occurrence of an event depends on its member outcomes that take place. The collection of all possible events is called the sample space , usually denoted by Ω . Thus, an event is a subset of the sample space, whose elements are called the sample points . Definition 2.3 A sample space that contains a finite or a countable collection of sample points is referred to as a discrete sample space . However, a sample space that contains an uncountable collection of sample points is referred to as a continuous sample space . 52 Probability, Statistics, Stochastic Processes Example 2.1 The following are examples of finite and infinite discrete sample spaces: i.
- Enrico Zio(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
141 Basics of Probability Theory for Applications to Reliability and Risk Analysis 4.1 Definitions In probabilistic terminology an experiment E is defined as a process whose outcome is a priori unknown to the analyst. The possible outcomes are all a priori known and classified but which one will occur is unknown at the time the experiment is performed. This definition is consistent with the Bayesian view of Probability according to which the outcome of an experiment may be deterministic, but at the moment unknown (e.g. the current number of sons of a friend with which one has lost contact long time ago) as well as stochastic (e.g. the result of a dice To each experiment E is associated a sample space Q , which represents the set of all possible outcomes of E . The sample space can be discrete finite (e.g. for an experiment of a coin or dice toss), countably infinite (e.g. the number of persons crossing the street in a given period of time: in principle, it could be infinite and yet be counted) or continuous (e.g. the value of the dollar currency in the year 3012). An event E is then a group of possible outcomes of the experiment E , i.e. a subset of 0. In particular, each possible outcome represents an (elementary) event itself, being a subset of R . Further, the null set 0 and the sample space R can also be considered events. - To each event E is possible to associate its complementary event E , constituted by all possible outcomes in R which do not belong to E . We say that event E occurs when the outcome of the experiment E is one of the elements of E . toss). 21 22 4 Basic of Probability Theory for Applications to Reliability and Risk Analysis 4.2 Boolean logic operations In the logic of certainty (Boolean logic), an event can either occur or not occur. Thus, it is represented by a statement, or proposition which can only be either true or false, and at a certain point in time, after the experiment is performed, the analyst will know its actual state.
eBook - PDFProbability and Statistics for STEM
A Course in One Semester
- E.N. Barron, J.G. Del Greco, E. N. Barron, J. G. Del Greco(Authors)
- 2022(Publication Date)
- Springer(Publisher)
1 C H A P T E R 1 Probability There are two kinds of events which occur: deterministic and stochastic (or random as a syn- onym). Deterministic phenomena are such that the same inputs always gives the exact same outputs. Newtonian physics deals with deterministic phenomena, but even real-world science is subject to random effects. Engineering design usually has a criteria which is to be met with 95% certainty because 100% certainty is impossible and too expensive. Think of designing an elevator in a high-rise building. Does the engineer know with certainty how many people will get on the elevator? What happens to the components of the elevator as they age? Do we know with certainty when they will collapse? These are examples of random events and we need a way to quantify them. Statistics is based on Probability which is a mathematical theory used to make sense out of random events and phenomena. In this chapter we will cover the basic concepts and techniques of Probability we will use throughout this book. 1.1 THE BASICS Every experiment whether it is designed by some experimenter or not, results in a possible outcome. If you deal five cards from a deck of cards, the hand you get is an outcome. Definition 1.1 The sample space is the set S of all possible outcomes of an experiment. S could be a finite set (like the number of all possible five card hands), a countably infinite set (like f0; 1; 2; 3 : : : g in a count of the number of users logging on to a computer system), or a continuum (like an interval OEa; b , like selecting a random number from a to b). Definition 1.2 An event A is any subset of S; A S: The set S is also called the sure event. The empty set ; is called the impossible event. The class of all possible events is denoted by F D fA j A S g: If S is a finite set with N elements, we write jS j D N and then the number of possible events is 2 N .
eBook - ePub- Giri(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
Section 2.1.1 we evaluated probabilities of events by the direct application of the definition of Probability, and this involved direct enumeration of the total number of equally likely cases and the number of cases favorable to an event. As problems grow in complexity the difficulty of direct enumeration of these cases also grows, and the computation of probabilities by direct application of the definition gets more involved. We shall now develop some theorems in Probability with a view to avoiding these complications.A theorem in Probability aims at deducing an algebraic relationship between probabilities of various related events, such that given the probabilities of some of these events, the probabilities of some other events that can be explained by them in some manner can be evaluated. That this is possible can be seen from the following simple example: If the Probability that an event A will happen is p, the Probability that the event A will not happen, usually represented as the eventA ¯, is 1 − p, whatever the event A may be. When the Probability of A is given, the Probability ofA ¯can be deduced immediately.Our general aim will be to develop a calculus of Probability based on theorems on Probability. The general logical scheme of the calculus of Probability is as follows: Given the probabilities of a set S of events for a set of random experiment E 1 , E 2 ,…, and given another random experiment E *, which is related to E 1 , E 2 ,… in a known manner, we should be able to make statements about the probabilities of specified events for E *, which can be explained in terms of the set S of events for E 1 , E 2 ,… in a certain sense.Two theorems are of fundamental importance in developing a calculus of Probability: the theorem of total Probability and the theorem of compound Probability. The statements of these theorems involve some basic concepts, such as mutually exclusive events, compound events, conditional Probability of an event given another event, and mutually independent events. Of these, we have already defined mutually exclusive events. The definitions of the remaining concepts appear below.Throughout we denote an event A by the symbol (A) and express the Probability of (A) as P(A). Consider an event A in conjunction with another event B
eBook - PDFRandom Phenomena
Fundamentals of Probability and Statistics for Engineers
- Babatunde A. Ogunnaike(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
1 Kingman, J.F.C. and S.J. Taylor. (1966). Introduction to the Theory of Measure and Probability , Cambridge University Press. Fundamentals of Probability Theory 69 3.3 Probability We are now in a position to discuss how to use the machinery we have assembled above to determine the Probability of any particular event A . 3.3.1 The Calculus of Probability Once the sample space Ω for any random experiment has been specified and the events (subsets of the sample space) identified, the following is the procedure for determining the Probability of any event A , based on the im-portant property that elementary events are mutually exclusive : • Assign probabilities to all the elementary events in Ω; • Determine the Probability of compound events from the prob-ability of the elementary events making up the compound event of interest. The procedure is particularly straightforward to illustrate for discrete sample spaces with a countable number of elements. For example, if Ω = { d 1 ,d 2 ,...d N } consists of N outcomes, then there are N elementary events, E i = { d i } . To each of these elementary events, we assign the Probability p i (we will discuss how shortly) subject to the constraint ∑ N i p i = 1. Now, if A = { d 1 ,d 2 ,d 4 } (3.24) and if P ( A ) represents the Probability of event A occurring, then, P ( A ) = p 1 + p 2 + p 4 (3.25) and for B = { d 3 ,d 5 ,...,d N } (3.26) P ( B ) = 1 − p 1 − p 2 − p 4 (3.27) The following examples illustrate how probabilities p i may be assigned to elementary events. Example 3.8 ASSIGNMENTS FOR EQUIPROBABLE OUT-COMES The experiment of tossing a coin 3 times and recording the observed number of heads and tails was considered in Examples 3.1 and 3.2. There the sample space was obtained in Eq (3.1) as: Ω = { HHH,HHT,HTH,THH,HTT,THT,TTH,TTT } , (3.28) 70 Random Phenomena a set with 8 elements that comprise all the possible outcomes of the ex-periment. Several events associated with this experiment were identified in Example 3.2.
eBook - PDF- Hans-Otto Georgii(Author)
- 2012(Publication Date)
- De Gruyter(Publisher)
A comprehensive and very stimulating historical and philosophical discussion of the notion of Probability can be found in Gigerenzer et al. [23]. Fortunately, the validity of the mathematical statements about a Probability model does not depend on its interpretation. The value of mathematics is not limited by the narrowness of human thought. This, however, should not to be misunderstood to mean that mathematics can take place in an ‘ivory tower’. Stochastics thrives on the interaction with the real world. 1.2 Properties and Construction of Probability Measures What are the implications of the assumption (A) of -additivity? We start with some elementary consequences. (1.11) Theorem. Probability rules. Every Probability measure P on an event space .; F / has the following properties, for arbitrary events A;B;A 1 ;A 2 ;::: 2 F . (a) P. ¿ / D 0 . (b) Finite additivity. P.A [ B/ C P.A B/ D P.A/ C P.B/ , and so in particular P.A/ C P.A c / D 1 . (c) Monotonicity. If A B then P.A/ P.B/ . (d) -Subadditivity. P S i 1 A i P i 1 P.A i / . (e) -Continuity. If either A n A or A n # A (i.e., the A n are either increasing with union A , or decreasing with intersection A ), then P.A n / ! n !1 P.A/ . Section 1.2 Properties and Construction of Probability Measures 15 Proof. (a) Since the empty set is disjoint from itself, the sequence ¿ ; ¿ ;::: consists of pairwise disjoint events. In this extreme case, -additivity (A) thus gives P. ¿ / D P.
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