7 Key excerpts on "Geometric Probability"

• 

  eBook - ePub

  Philosophical Foundations of Probability Theory

  • Roy Weatherford(Author)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  103 This may well be so. But evidently not everyone falls in these exalted categories, for consider these recent definitions:

  Probability. The ratio of the number of ways in which an event can occur in a specified form to the total number of ways in which the event can occur.104

  Probability, Mathematical. If an event can happen in a ways and fail in b ways, and, except for the numerical difference between a and b, is as likely to happen as to fail, the mathematical probability of its happening is a/(a + b) and of its failing, b/(a + b).105

  The Classical definition lives! And as long as human beings continue to face situations where the outcome is unknown but the alternatives are all felt to have an equal chance of occurrence – as long, in short, as we continue to gamble at dice and cards – the Classical Theory of Probability will continue to be the working theory of the ordinary person.

  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)

  k birth dates (the total numbers of possible outcomes) is

  The number

  nB

  can be calculated as the number of permutations of 365 elements taken k at a time (refer to Appendix 2 ):

  The resulting formulas for the probability P (B ) are given by equations

  or

  If k = 23, then P (B ) = 0.493, and P (A ) = 1 − 0.493 = 0.507.

  This number represents the probability of occurrence of two or more earthquakes on the same day (if 23 quakes occurred during the year), as well as the probability that two or more participants of a soccer match (including referee) share the same birth date.

  GEOMETRIC DEFINITION OF PROBABILITY

  Let us suppose that G is some domain in the Euclidean space of any finite dimension (line, plane, etc.). Consider a stochastic experiment that consists in random choice of a point from the domain G . Assume that for any A ⊂G the probability of event “the chosen point belongs to A ” does not depend on location inside G and is proportional to the measure of A . How can one define the probability that a point randomly chosen from domain G belongs to region A ?

  First of all, one needs to compose a probabilistic space that matches this experiment. For that purpose, it is possible to consider the set of all points from G as Ω and the collection of subsets of G having a measure in the primary Euclidean space as ϕ. The probability P (A ) to choose a point inside A is completely determined by the stated conditions. Namely, according to our conditions P (A ) = C μ(A ), where μ(A ) is the measure of A , and C is a proportionality coefficient. In particular, P (G ) = P (Ω) = C μ(G ) = 1 is the probability of sure event. Therefore, C = 1/μ(G). Taking this value for C , one yields:

  (4.12)

  Equation 4.12 is called the geometric definition of probability

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Chance and the Sovereignty of God

  A God-Centered Approach to Probability and Random Events

  • Vern S. Poythress(Author)
  • 2014(Publication Date)
  • Crossway

    (Publisher)

  The application of mathematical probability to many distinct phenomena is one instance of the principle of the one and the many. A general mathematical result about mathematical probability is a single result. It shows unity. It also has applications to many physical situations. The many situations show the diversity. The many situations go together with the one general result. At the same time, the general result is motivated by the long-range purpose of applications to the many physical situations. Human beings understand the meaning of the general result by referring to particular illustrations that apply it. Conversely, the particular illustrations gain meaning through being seen as embodiments of one general principle. The relations between the one and the many go back, as usual, to a foundation in God, who is the original one God in three persons.

  For example, suppose we know that we have a situation where one of two mutually exclusive events A or B can take place, and where the probability of the event A is 1/2. Then the probability of the other event B is also 1/2 (because the total probability for both together must be 1). The general principle says that the probabilities must add to 1: 1/2 + 1/2 = 1. This general principle applies to the flip of the coin, which can come up heads or tails. It also applies to the roll of a die, which can come up odd (1, 3, or 5) or even (2, 4, or 6). And it applies to many other physical situations in the world. God ordains the consistency between the general principle, the one, and the particular applications, the many.

  BASIC CONSTITUENTS FOR A MATHEMATICAL MODEL

  The transition from physical situations to a mathematical treatment takes place by producing a kind of mathematical model for the physical situations. The model strips out all the particulars about coins, dice, cards, and slot machines. Instead, we start with an abstract set S, the set (collection) of all possible outcomes for a trial of some kind.1 For example, for a die roll, the outcomes are 1 through 6, so the set S has members 1, 2, 3, 4, 5, and 6. The usual notation for writing the members of a set S

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Statistical Techniques in Geographical Analysis

  • Dennis Wheeler, Gareth Shaw, Stewart Barr(Authors)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  5

  Probability and Probability Distributions

  5.1  Introduction

  PROBABILITY CAN BE DEFINED in both the general and the mathematical senses. We might, for example, say ‘it is probable that you will understand this book’. From this we could infer that the reader has a greater chance of understanding the text than of being bewildered by it. But such an expression can be interpreted in only the vaguest of terms, and scientists prefer to use the word in a more rigorous fashion, attaching some numerical value to the probability of an event. This numerical probability can be expressed in either of two ways – on an absolute scale of zero to one, or on a percentage scale of zero to 100. Both are widely used.

  It is possible to think of some events to which numerical probability values can be readily attached. Some events – the daily setting of the sun, or gradual erosion of land surfaces – can be thought of being absolutely certain. Such absolute certainties have a probability of 1·0, or 100 per cent. Conversely, for impossible situations the probability is 0·0 or 0 per cent. However, most events are by no means impossible or certain, and the probabilities of their occurrence can be thought of as lying at some point along a probability spectrum between 0·0 and 1·0 (or 0 to 100 per cent). Figure 5.1 illustrates the position for some simple, if non-geographical, events.

  Figure 5.1    Probability scale from 0 to 1, showing the likelihood of some everyday events

  5.2  Assessment of Probability Values

  Logical reasoning can derive the numerical probabilities of inevitable, of impossible, and of some other events between those extremes, but alternative methods are needed in the less obvious cases. The French mathematician Pierre Simon de Laplace (1749–1327) was the first to define and solve the problem algebraically. If the numerical probability of an event x is denoted by p(x)

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Introduction to Statistics for Forensic Scientists

  • David Lucy(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  3

  Probability

  Even though there is no complete agreement about the fundamental nature of probability amongst statisticians and probability theorists, there is a body of elementary probability theory about which all agree. The fundamental principle is that the probability for any event is between 0 and 1 inclusive, that is, for any event A 0 ≤ Pr(A ) ≤ 1 where Pr stands for probability, and the element in the parentheses is the event under consideration. This is the first law of probability , sometimes known as the convexity rule , and implies occurs with probability of event which cannot happen, an event which occurs with probability 1 is an event which must happen, and events which occur with probabilities between 0 and 1 are subject to some degree of uncertainty.

  3.1 Aleatory probability

  Aleatory† probabilities are the calculation of probabilities where those probabilities can be notionally deduced from the physical nature of the system generating the uncertainty in outcome with which we are concerned. Such systems include fair coins, fair dice and random drawing from packs of cards. Many of the basic ideas of probability theory are derived from, and can best be described by these simple randomization devices.

  One throw of a six-sided die

  If I have a fair die the probability of throwing a six from a single throw is 1 in 6 or 1/6 = 0.17 or 17%. There are six faces on a fair dice, numbered 1, …, 6, any of which in a single throw is equally likely to end face up, and is certain that the die must end its roll with one face uppermost. The event in which we are interested is the outcome of a six facing uppermost. Stating this intuition as an equation:

  For the dice there is one way in which a six can be thrown, and six ways in which a dice can land, therefore Pr(A ) = 1/6, or 17%.

  If the event of interest is the probability of not throwing a six then we know there are five ways in which a fair six-sided die can be thrown, and a six not land uppermost. There are six equally likely ways in which a fair die can land, and exactly one face must land uppermost. Therefore Pr(B ) = 5/6 = 0.83, where B

  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)

  A ). Mathematically, a scale ranging from 0 to 1 is used to evaluate the likelihood of occurrence of an event. If an event is very likely to occur, it is assigned a probability close to 1. If an event is very unlikely to occur, it is assigned a probability close to 0. It follows, then, that an event that is ‘certain’ to occur has a probability of 1, while an event that is ‘impossible’ has a probability of 0. The probability of the event that the sun will rise tomorrow is 1. The probability of the event that a tossed coin will not land anywhere (stays in the air) is 0. In practical applications, probabilities are often converted to percentages, with the possible values ranging from 0% to 100%, and are frequently referred to as chances. For example, a weather forecaster may say that, ‘The chance of showers tomorrow is 80%,’ meaning that the probability of rain tomorrow is 0.8.

  There are three kinds of probabilities: classical, empirical and subjective.

  Classical probability is calculated from the knowledge of the sample space and an event from a random experiment. It is so named because it was the first type of probability studied by mathematicians in the 17th century. As we discussed in Section 3.2 , the probability of an event, A , can be calculated from the total number of outcomes in a sample space, n , and the number of ways that event A can occur, f .

  In other words, f is the number of outcomes in event A , whereas n is the number of total outcomes in the entire sample space. Equation 3.7 assumes the total number of outcomes, n , is equally likely; that is, they all have exactly the same probability of occurring.

  Example 3.18. Two dice, one red and the other green, are rolled. What is the probability of event A , defined as having the number of dots totalling 7, occurring? All of the 36 outcomes in this sample space are listed, with the event A defined in boldface :

  A = {1–6, 2–5, 3–4, 4–3, 5–2, 6–1}

  Since n = 36 and f = 6, .

  Empirical probabilities are based on experiments for which the possible outcomes and the number of outcomes favouring an event are not known exactly, but generally have been observed. If an experiment is repeated n times and f out of the n trials favours event B

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Statistics for the Behavioural Sciences

  An Introduction to Frequentist and Bayesian Approaches

  • Riccardo Russo(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  Notice, therefore that probability, as being conceptualised as a relative frequency of the occurrences of a specific event A relative to the occurrences of all the events from the sample space where A belongs to (e.g., take, as above, the ratio of the number of “1” observed divided the total number of events “1–6” been seen when the die has been rolled potentially an infinite number of times), can only be applied to collectives and not to single situations. The reason being that the long-term relative frequency is a property of all the members of the collective and not of an individual event. Hence, probability, in the above sense, is objectively related to the repeated occurrence of events in the real world. The total number of observations is the reference class or collective.

  The analytic view of probability

  According to the analytic view of probability, if a sample space consists of n(S) equally likely events, then the probability of occurrence of the event E is given by:

  P( E )   =   n ( E ) n ( S )

  where n(E) is the number of times in which the event E occurs, and n(S) is the number of simple events in the sample space S. Notice that it is assumed that events are equally likely!

  Thus, for example, the probability of obtaining an even number when rolling a fair die is given by:

  P(Even   number) = n ( Even   number ) n ( S ) = 3 6 = 0.5

  where n(Even number) = 3 (i.e., simple events “2”, “4”, and “6”), and n(S) = 6 (i.e., the sample space of all 6 possible numerical outcomes).

  Subjective view of probability

  There is also a third way to consider probabilities. This is the subjective view, and corresponds to the subjective or personal belief that people hold about the likelihood of occurrence of a particular event. Statements like “it is very likely that it will rain this morning”, “it is unlikely that the football team I support is going to win the league this year”, and “there is a 60% probability that his theory is true” capture the subjective appraisal of the probability of occurrence of an event. The subjective view of probability is at the core of the Bayesian approach to inferential statistics.

  Neither the Frequentist nor the analytic approaches to probability can handle statements like (assuming Atlantis exists): “What is the probability that candidate A will win over candidate B in the election of the president of Atlantis next year?”. This is a single unique event for which there is not a collective to be used to calculate a relative frequency, nor there is any analytical way to determine its probability. It is not possible to have a potential infinite number of repetitions of this single event to be used to calculate the relative frequency of the wins of candidate A out of the total number of elections. If you say, for instance, that A has a 40% chance for to win, this represents your personal belief.

  Sign up to read

  Learn more about book