8 Key excerpts on "Events (Probability)"

• 

  eBook - ePub

  Statistics for Business

  • Perumal Mariappan(Author)
  • 2019(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  8 Probability 8.1      Introduction

  The concept of probability was introduced late in the seventeenth century. This concept was introduced in problems relating to games of probability (i.e., tossing a coin, playing cards). But the probability concept is now used in almost all areas of study such as economics, statistics, industry, engineering, and business. Probability is related to the study of events that are going to happen or not.

  Before going further, let’s define some of the basic terms that are going to be used in the definition of probability. 8.2      Definitions for Certain Key Terms 8.2.1      Experiment An experiment means an activity or measurement that result in an outcome.

  Example:

  Tossing a single coin for 50 times. 8.2.2      Sample Space Sample space refers to the collection of all possible events of an experiment, denoted by S.

  Example:

  In a coin-tossing experiment, the sample space should contain the possible outcomes of a head (H) or a tail (T); S = {H, T} 8.2.3      Event Event means one or more of the possible outcomes of an experiment; it is a subset of a sample space.

  Example:

  In throwing a dice, S = {1, 2, 3, 4, 5, 6} contains the face 1 is an event. 8.2.4      Equally Likely Events In a sample space containing at least 2 events, the chance of the occurrence of each of the event is equal.

  Example:

  In a coin-tossing experiment, having a head or tail in a trial is equal to ½ each or 50%. 8.2.5      Mutually Exclusive Events Events are said to be mutually exclusive if the outcome is only 1 element at a time. There is no chance that 2 or more events happen together. Alternatively, it is called an ‘incompatible event’.

  Example:

  In a coin-tossing experiment, we can have either head or tail as an outcome. Clearly the occurrence of head prevents the occurrence of the tail, which implies that the 2 events are said to be mutually exclusive.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Statistics and Probability

  • Britannica Educational Publishing, Nicholas Faulkner(Authors)
  • 2017(Publication Date)
  • Britannica Educational Publishing

    (Publisher)

  This text contains a description of the important mathematical concepts of probability theory, illustrated by some of the applications that have stimulated their development. Since applications inevitably involve simplifying assumptions that focus on some features of a problem at the expense of others, it is advantageous to begin by thinking about simple experiments, such as tossing a coin or rolling dice, and later to see how these apparently frivolous investigations relate to important scientific questions.

  Experiments, Sample Space, Events, and Equally Likely Probabilities Applications of Simple Probability Experiments

  The fundamental ingredient of probability theory is an experiment that can be repeated, at least hypothetically, under essentially identical conditions and that may lead to different outcomes on different trials. The set of all possible outcomes of an experiment is called a “sample space.” The experiment of tossing a coin once results in a sample space with two possible outcomes, “heads” and “tails.” Tossing two dice has a sample space with 36 possible outcomes, each of which can be identified with an ordered pair (i, j ), where i and j assume one of the values 1, 2, 3, 4, 5, 6 and denote the faces showing on the individual dice. It is important to think of the dice as identifiable (say by a difference in colour), so that the outcome (1, 2) is different from (2, 1). An “event” is a well-defined subset of the sample space. For example, the event “the sum of the faces showing on the two dice equals six” consists of the five outcomes (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).

  A third example is to draw n balls from an urn containing balls of various colours. A generic outcome to this experiment is an n -tuple, where the i th entry specifies the colour of the ball obtained on the ith draw (i = 1, 2,…, n

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Probability in Petroleum and Environmental Engineering

  • George V Chilingar, Leonid F. Khilyuk, Herman H. Reike(Authors)
  • 2012(Publication Date)
  • Gulf Publishing Company

    (Publisher)

  Probability theory deals mostly with stochastic experiments and random events. For random events, before the beginning of an experiment, one can assert only the possibility of their occurrence. There are a lot of examples of events for which it is possible to guess how often they occur in a series of same experiments. For such events one can introduce a certain quantitative measure to characterize the chances of their occurrence in a particular experiment. The probability of a random event is such a measure in the probability theory. This is a central concept of the theory that is discussed here.

  Consider a certain stochastic experiment and some algebra ϕ of its events. Assume that for every A ∈ ϕ there is some real number P (A ) for which the following conditions (axioms) are satisfied for any mutually exclusive events A and B :

  1. Standardization axiom .

  (4.1)

  2. Nonnegativity axiom .

  (4.2)

  3. Additivity axiom .

  (4.3)

  Conditions 4.1–4.3, defining number P (A ) in a certain way, are called the axioms of probability. From a mathematical point of view, probability is a function P (satisfying the conditions 1, 2, 3) that is defined over an algebra of events ϕ. The probability of event A ∈ϕ is the value of this function at the “point” A .

  Remark 4.1. There is one additional property of probability that is used mostly in advanced probabilistic studies. Let be a decreasing sequence of events. Introduce the event . This additional condition is:

  4. Continuity axiom .

  (4.4)

  .

  Axioms 4.1–4.4 express general desirable properties of probabilities of any experiment. For each specific experiment, probability has to be defined particularly, keeping in mind the restrictions imposed by these axioms.

  Definition 4.3. A triple symbol , where Ω is the space of elementary events of a given experiment, ϕ is the algebra of events, and P is the, where ϕis the space of elementary events of a given experiment, ϕ is the algebra of events, and P is the probability defined for every A ∈ϕ, is called the probabilistic space of a given stochastic experiment.

  The three mathematical symbols enumerated in the braces represent a mathematical model of a given stochastic experiment. Space of elementary events Ω together with algebra ϕ define all possible events of the experiment through elementary events ω. The function P prescribes particular probabilities associated with every event A of the experiment. Thus, all possible outcomes of a given experiment and their theoretical frequencies of occurrence are described formally by the triple symbol

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Business Statistics with Solutions in R

  • Mustapha Abiodun Akinkunmi(Author)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  4 Basic Probability Concepts In this chapter, we will discuss some basic concepts of probability, solving the problem of probability using Venn diagrams. The axioms and rules of probability will be discussed and will be extended to conditional probabilities. Practical examples with R code will be used for illustration. 4.1 Experiment, Outcome, and Sample Space As we have mentioned before, it is important to understand the definitions of certain terms to be able to use them successfully. So, the first step in gaining an understanding of probability is to learn the terminology and the rest will be a lot simpler. 4.1.1 Experiment Experiment is a measurement process that produces quantifiable results. Some typical examples of an experiment are: the tossing of a die, tossing of a coin, playing of cards, measuring weight of students, and recording growth of plants. 4.1.2 Outcome Outcome is a single result from a measurement. Examples of outcomes are: getting a sum of 9 in the tossing of two dice, turning up of heads in the toss of a coin, selecting a spade from a deck of cards, and getting a weight above a certain threshold (say 50 kg). 4.1.3 Sample Space The sample space is the set of all possible outcomes from an experiment and is denoted with S. The sample space of tossing a die is S = 1, 2, 3, 4, 5, 6 ; the sample space of tossing a coin is denoted as S = H, T. In addition, the sample space for tossing two dice. is S = 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 4, 1, 4, 2,[--=PLGO-SEP. ARATOR=--]4, 3, 4, 4, 4, 5, 4, 6, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6 4.2 Elementary Events Any subset of a sample set, empty set, and whole set inclusive is called an event. An elementary event is an event with a single element taken from a sample space and it is denoted as E

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introductory Probability and Statistics

  Applications for Forestry and Natural Sciences (Revised Edition)

  • Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
  • 2019(Publication Date)
  • CAB International

    (Publisher)

  3 Probability The Foundation of Statistics We use statistical information every day to qualify statements and to help us make decisions. For example, we may hear statements like: • There is an 80% chance of rain today. • The odds are one in 13 million that you will win the lottery. Or we may be confronted with questions like: • What is the likelihood of receiving an A on the first exam in this course? • What is the chance that the Vancouver Canucks will win the next Stanley Cup? Statistical inference, the generalization from a sample to a population, involves drawing a conclusion about a population on the basis of available, but incomplete, information. Hence, statistical inference involves a certain amount of uncertainty, and statisticians should not base decisions on statistical inference unless the risk of uncertainty can be reduced to a tolerable minimum. Problems involving ‘uncertainty’, ‘chance’, ‘likelihood’, ‘odds’ and other such factors require an understanding and application of the theory of probability. Probability is the branch of mathematics that incorporates the most important set of concepts used in the field of statistics. The purpose of this chapter is to introduce the basic theories of probability that are required to appreciate and understand many of the concepts of statistical inference. 3.1    Sample Space and Events In statistics, we define an experiment as a process that produces some data. In Chapter 1, we described an experiment to study the effects of seeding date and seedbed preparation on germination. A wood scientist could be interested in studying the effect of temperature and applied pressure on the strength properties of plywood. Experi ments such as tossing a coin, rolling a dice, or drawing a card from an ordinary (52 cards) deck of cards will also produce some data

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Risk Analysis in Building Fire Safety Engineering

  • A. Hasofer, V.R. Beck, I.D. Bennetts(Authors)
  • 2006(Publication Date)
  • Routledge

    (Publisher)

  S.

  For example, if we light a fire in a compartment, we could consider three possible outcomes: a smouldering fire, a flaming fire and a flashover fire. In that case, the sample space of the experiment consists of three elements. If we repeat the experiment often enough we may be able to state that we get a smouldering fire 30% of the time, a flaming fire 50% of the time and a flashover fire 20% of the time. In this way, we can attach probabilities to the elements of the sample space. Probabilities are expressed as fractions, not percentages. Here the three outcomes will be said to have probabilities of 0.3, 0.5 and 0.2, respectively.

  Sometimes there is an infinity of possible outcomes. For example, we might be measuring the maximum temperature reached in a compartment. It might be any number between, say, 500◦ C and 1500◦ C. The sample space in this case is the interval (500, 1500).

  3.2.1 Events

  An event is a subset of the sample space. For example, we may be interested in fires where the maximum temperature reached is between 1000◦ C and 1200◦ C. The interval (1000, 1200), a subset of the sample space (500, 1500), is called an event.

  Definition 3.2.1. An event that consists of only one point of the sample space, i.e. just one outcome, is called an elementary event.

  Since events are sets, they obey the laws of set operations. Let A and B be two events. Then:

  1. The complement of A, AC , also called not A, is the set of elements of S not in A.

  2. The union of A and B, A ∪ B, also called A or B, is the set of elements of S in either A or B (i.e. the symbol ∪ stands for or).

  3. The intersection of A and B, A ∩ B, also called A and B, is the set of elements of S in both A and B (i.e. the symbol ∩ stands for and).

  Clearly, A ∪ B and B ∩ represent the same event. Similarly for A ∪ B and B ∩ A

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Mathematical Finance

  Core Theory, Problems and Statistical Algorithms

  • Nikolai Dokuchaev(Author)
  • 2007(Publication Date)
  • Routledge

    (Publisher)

  1Review of probability theory

  In probability theory based on Kolmogorov’s probability axioms, the model of randomness is the following. It is assumed that there exists a set Ω, and it is assumed that subsets A ⊆ Ω are random events. Some value P(A) ∈ [0, 1] is attached to any event as the probability of an event, and P(Ω)=1. To make this model valid, some axioms about possible classes of events are accepted such that the expectation can be interpreted as an integral.

  1.1 Measure space and probability space

  σ-algebra of events

  Let Ω be a non-empty set. We denote by 2Ω the set of all subsets of Ω.

  Example 1.1 Let Ω={a, b}, then 2Ω ={∅, {a}, {b}, Ω}.

  Definition 1.2 A system of subsets Ƒ ᑕ 2

  Ω

  is called an algebra of subsets of Ω if

  (i)Ω ∈ Ƒ ;

  (ii) If A ∈ Ω then Ω\A ∈ Ƒ :

  (iii) If A

  1,

  A

  2

  ,..., A

  n

  ∈ Ƒ, then ∈ Ƒ .

  Note that (i) and (ii) imply that the empty set ∅ always belongs to an algebra.

  Definition 1.3 A system of subsets Ƒ ᑕ 2

  Ω

  is called a σ-algebra of subsets of Ω if

  (i) It is an algebra of subsets;

  (ii) If A

  1,

  A

  2

  ,... ∈ Ƒ (i.e ᑕ Ƒ), then A

  i

  ∈ Ƒ .

  Definition 1.4 Let Ω be a set, let Ƒ be a σ-algebra of subsets, and let : Ƒ → [0, + ∞ ]be a mapping.

  (i) We said that µ is a σ-additive measure if µ = for any A

  1,

  A

  2

  ,... ∈ Ƒ such that A

  i

  ∩ A

  j

  = Ø if i ≠ j . In that case, the triplet ( Ω, Ƒ, ) is said to be a measure space.

  (ii) If µ(Ω)<+∞, then the measure µ is said to be finite.

  (iii) If µ(Ω)=1, then the measure µ is said to be a probability measure.

  To make notations more visible, we shall use the symbol P for the probability measures.

  Definition 1.5 Consider a measure space (Ω, Ƒ, µ) . Assume that some property holds for all (ω ∈ Ω

  1

  , where Ω

  1

  ∈ Ƒ is such that μ(Ω\Ω1 )=0. We say that this property holds a.e. (almost everywhere). In the case of a probability measure, we say that this property holds with probability 1, or a.s. (almost surely).

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Basic Statistical Techniques for Medical and Other Professionals

  A Course in Statistics to Assist in Interpreting Numerical Data

  • David Smith(Author)
  • 2021(Publication Date)
  • Productivity Press

    (Publisher)

  Figure 2.2 .

  Figure 2.2

  Mutually exclusive events

  The previous theorem does not apply here since the probability of both event A and event B taking place is zero. It follows that the probability of drawing either a heart or a black card (event A or event B) is

  P a + P b

  and for multiple events, the probability of observing either event A, or event B, or event C, or event D, etc., is

  P a + P b + P c + P d e t c

  In the playing card example Pa = 0.25 Pb = 0.5 and therefore the probability of drawing either a heart or a black card is 0.25 + 0.5 = 0.75.

  Exercise 1 Manipulating Probabilities Assume that:

  Having dark hair is 80% likely. Having blue eyes is 25% likely. Being bald is 5% likely. Being taller than 1.8 m is 50% likely.

  What is the probability of the following:

  1. Having dark hair and blue eyes?
  2. Having dark hair or blue eyes?
  3. Having dark hair and being bald?
  4. Having dark hair and blue eyes and being taller than 1.8 m?
  5. Having dark hair or blue eyes or being taller than 1.8 m?
  6. Having dark hair and blue eyes, or being bald?
  7. Having neither blue eyes nor not being bald?

  Conditional Probabilities

  The following example illustrates a state of affairs in which the probability of an event is conditional on the probability of some other factor. Assume that a person, at random, has a 1% probability of suffering from a particular form of cancer (in other words 1% of the population are known to suffer). Assume, as is often the case, that there is a test to determine if the cancer is present but that the test is not perfect. Like many tests, it may return a false positive despite the subject not suffering from the condition and it might also return a false negative in that it fails to detect a real case. The picture is summarised in the following table with the assumption that there is a 10% chance of a false result (be it positive or negative).

  Sign up to read

  Learn more about book