Geometric Probability

7 Key excerpts on "Geometric Probability"

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  Mathematics of Chance

  • Jirí Andel(Author)
  • 2009(Publication Date)
  • Wiley-Interscience

    (Publisher)

  1.3 Geometric Probability Sometimes the space R is a Borel subset of the Euclidean space I & with a positive and finite Lebesgue measure X(R). If it is ensured that P(A) = X(A)/X(R), for every Borel set A c R, then no elementary event w E R is preferred and the probability of any Borel set A C R is proportional only to its Lebesgue measure. The measure P , called geometricprobability, was applied in calculations for different experiments performed as early as the eighteenth century. At that time Geometric Probability was the only way to extend the calculation of the classical probability to some noncountable spaces 0. Consider an idealized watch with a classical face such that the second hand moves continuously. If we want to look at it, we can ask about the probability that the second hand points to a place between 1 and 3 . This ark A forms one-sixth of the whole circle R. So the abovementioned probability is i. The problem of meeting is usually solved by means of Geometric Probability. Two friends, say, X and Y, agreed to meet at a given place between noon and I p.m. Say that X comes to the appointed meeting place randomly during this interval and independentlyof Y, who also comes randomly. Each of them will wait for 10 minutes, but not after 1 p.m. Calculate the probability that the friends meet. This situation is described in Fig. 1.1. The moment of arrival of Mr. X is rep- resented as a randomly chosen point in segment AB and similarly the moment of arrival of Mr. Y a randomly chosen point in segment AD. The scale on both axes is considered in minutes. The friends meet if and only if the point (z, y) falls into polygon AEFCGH. The area of square ABCD is 602. The sum of the areas of triangles EBF and GDH is 502. Then the area of the polygon AEFCGH makes 602 - 50’ = 1100. The probability that X and Y meet is - 0.306. p = --1100 602 One problem solved using geometric probabilities showed strange results. A circle K is given and a chord is randomly chosen in it.

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  Statistical Methods Of Geophysical Data Processing

  • Vladimir Troyan, Yurii Kiselev(Authors)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  By using the properties of probability pairwise of not intersected events, we get the formula P ( A ) = m X k =1 P ( E i k ) . Hence, in case of the finite experiment the probability of any event is determined by probabilities of elementary events. For example, taking into account a symmetry of experiment, it is possible to establish a priori, that the elementary events have an equal probability (the probability of the elementary event is equal 1 /n , where n is a number of equally possible outcomes), and the probability of event A is calculated as the relation of number of the favorable outcomes m to number of equally possible outcomes n : P ( A ) = m n . 6 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING Usually calculation of equally possible and favorable outcomes is carried out by the combinatorial methods Such definition is possible to subject to criticism on the ground that the notion equally possible actually means equality probability , and reasoning, thus, contains a vicious circle. 1.1.5 Geometrical definition of probability In case of experiments with infinite number of equally possible outcomes, when the effect(result) of the experiment can be connected with a point belonging to R m , the probabilities of some events can be defined geometrically as the relation of Euclidean volume (area, length) for the part of a figure to volume (area, length) of complete figure. An illustration of geometrical definition of probability is given on Fig. 1.5. Fig. 1.5 An illustration of geometrical definition of probability. 1.1.6 Exercises (1) Two bones are thrown. To find probability that the total number of pips on the happen to be facets is even, and on a facet even by one of bones will appear six pips. (2) In the container there are 10 identical devices marked with the numbers 1 , 2 , .

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  Theory of Probability

  • Boris V. Gnedenko(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  FIGURE 3. Geometrical interpretation or the solution for the rendezvous problem. 4    GEOMETRICAL PROBABILITY

  The “classical” definition of probability based on the consideration of a finite group of equally probable events was considered as insufficient even at the very beginning of the development of the theory of probability. Even at that time, particular examples led to some modification of this definition and to a formulation of a concept of probability applicable to situations in which the set of conceivable outcomes was infinite. Here, nevertheless, the notion of “equal likelihood”of definite events played a central role.

  The general problem which was posed, and which led to an extension of the concept of probability, can be formulated in the following way.

  Suppose, for example, that there exists a region G in a plane and that it contains another region g. A point is thrown at random onto the region G, and the probability that the point falls in the region g is under investigation. The expression “thrown at random onto the region G” means that the probability of the point’s falling in some part of the region G is proportional to its measure (i.e., its length, area, volume, etc.) and is independent of its location and its shape. Therefore, by definition,

  p =  

  mes   g

  mes   G

  is the probability that a point chosen at random in the region G fall within the region g.

  Let us now consider several examples.

  Example 1. The rendezvous problem. Two persons A and B have agreed to meet at a specified place between noon and 1 p. m. The one who arrives first waits 20 minutes for the other, after which he leaves. What is the probability of their meeting if their arrivals during this hour occur at random, and if the moments of their arrivals are independent3 ?

  SOLUTION. Let x and y denote the time of arrival of A and of B, respectively. In order for a rendezvous to occur, it is necessary and sufficient that |x – y| ⩽20. Let x and y be the Cartesian coordinates of a point on a plane and one minute be taken as a unit of distance. All the possible outcomes will be depicted by the points inside a square of side 60, and the outcomes favorable to a meeting by the points of the shaded region (Figure 4

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  Distributed Sensor Networks

  Sensor Networking and Applications (Volume Two)

  • S. Sitharama Iyengar, Richard R. Brooks, S. Sitharama Iyengar, Richard R. Brooks(Authors)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  125 6.1 Geometric Probability 6.1.1 Introduction In seeking to maximize the probability of target detection by sensors randomly distributed over some region, one might consider factors such as the numbers of sensors, the detective power of the sensor, and the way in which the sensors are spatially distributed throughout the region. Many of these factors have a strong probabilistic and geometric nature. In this chapter, we use techniques from geometric prob-ability and integral geometry to investigate coverage and detection problem for sensor fields. Methods of integral geometry have been used in computing coverage probabilities in heterogeneous sensor net-works by Lazos et al. [15] using many of the results in the classic work of Santaló [23]. Similarly, Rowe and Wettergren [22] used Geometric Probability methods to examine the reliability over time of ran-domly distributed heterogeneous sensor networks. This section outlines some techniques from Geometric Probability that can be used to examine the coverage and performance of randomly distributed heterogeneous fields consisting of a large number of sensors. The remainder of this chapter uses techniques from integral and stochastic geometry to 6 Application of Geometric Probability and Integral Geometry to Sensor Field Analysis 6.1 Geometric Probability ...................................................................... 125 Introduction • Spatial Poisson Processes • Sensor Coverage • Coverage Comparison Study • Monte Carlo Study of the Coverage Statistic • Simple Network Communications Application 6.2 Integral Geometry I ..........................................................................

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  Luck, Logic, and White Lies

  The Mathematics of Games

  • Jörg Bewersdorff(Author)
  • 2004(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  In general, one can find a probability for an arbitrary interval that depends only on the geometric “size” of the interval, by which we mean the relative size in relation to the entire inter-val. The addition law for such “geometric” probabilities then amounts to nothing more than adding lengths. But how is one to treat such geometric probabilities mathematically? For example, in Buffon’s needle problem, how can the assumptions about the equiprobabilities of angles and distances lead to the desired probability? Why, based on empirical experiment, is that probability equal to 2/7r = 0.6366 in the case of a stick of length equal to the distance between boards? Before we delve into the not-so-simple needle problem, we would like to consider a similar question, one that also goes back to Buffon: a coin of radius r is tossed, and it lands on a tiled floor, the tiles of which are squares of side a. What is the probability that the coin does not touch a line between tiles (we consider these lines to have zero width)? This problem is easier to the extent that only one geometric value is needed to describe the result of tossing the coin, namely, the point on the floor on which the center of the coin lands. Here the situation is the same for each tile (see Figure 7): the event that the coin does not touch a gap occurs when the center of the coin lies within a square centered on the tile whose sides, parallel to those of the tile, are at a distance the radius of Figure 7.1. The ways in which a coin can land on a tiled floor. Games of Chance 39 the coin r from those of the tile. Such a square exists only if the length a of the tiles is greater than the diameter 2 r of the coin. Since every point on the floor is an equiprobable landing site, the probability that no gap is touched by the coin is equal to the ratio of the area of the smaller square to that of the tile, that is, (a — 2r ) 2/ a 2.

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  Discrete Distributions in Engineering and the Applied Sciences

  • Rajan Chattamvelli, Ramalingam Shanmugam(Authors)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  65 C H A P T E R 4 Geometric Distribution After finishing the chapter, readers will be able to : : : • Understand geometric distributions. • Explore properties of geometric distribution. • Discuss arithmetico-geometric distribution. • Apply geometric distribution to practical problems. 4.1 DERIVATION Consider a sequence of independent Bernoulli trials with the same probability of success p. We observe the outcome of each trial, and either continues it if it is not a success, or stops it if it is a success. This means that if the first trial results in a success, we stop further trials. If not, we continue observing failures until the first success is observed. Let X denote the number of trials needed to get the first success. Naturally, X is a random variable that can theoretically take any value from 0 to 1. In summary, practical experiments that result in a geometric distribution can be characterized by the following properties. 1. The experiment consists of a series of IID Bernoulli trials. 2. The trials can be repeated independently without limit (as many times as necessary) under identical conditions. The outcome of one trial has no effect on the outcome of any other, including next trial. 3. The probability of success, p, remains the same from trial to trial until the experiment is over. 4. The random variable X denotes the number of trials needed to get the first success (q x 1 p), or the number of trials preceding the first success (q x p). If the probability of success is reasonably high, we expect the number of trials to get the first success to be a small number. This means that if p D 0:9, the number of trials needed is much less than if p D 0:5, in general. If a success is obtained after getting x failures, the probability is f .xI p/ D q x p by the independence of the trials. This is called the geometric 66 4. GEOMETRIC DISTRIBUTION distribution, denoted by GEO(p).

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  Introduction to Probability and Statistics

  • Giri(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  n, and

  ∑

  i = 1

  n

  P i

  = 1

  to the points s1 ,s2 ,…, sn , respectively, such that pi is the probability of the sample point si , i = 1,2,…, n, we can define the probability of an event A within the framework of the classical definition of probability as follows: Let SA be the subset of S defining the event A, and let SA consist of the points s

  A1

  ,s

  A2

  ,…, s

  A

  m

  , where the s

  A

  i

  are some of the points s1 ,s2 ,…, sn of the sample space S; the probability of A is defined as

  P

  ( A )

  =

  ∑

  i = 1

  m

  P

  A i

  .

  It can easily be verified that this extended definition of probability within the framework of the classical setup (based on the assumption that a sample space contains a finite number of points) does not disturb any of the theorems of probability we developed in the preceding sections. The basic steps in the proofs of these theorems remain unaltered.

  Generally speaking, the framework for the definition of classical probability theory can be summarized as consisting of a set S (called the sample space) and a family F of subsets of S (called the elementary events) whose measures (probabilities) are prescribed in advance. Finally, a set of rules is postulated whereby measures (or equivalently probabilities) of subsets of S (events) can be computed. The accepted rules for classical probability theory are as follows:

  1.  S consists of a finite number of points (elementary events).

  2.  Denoting by μ the measure (probability), we have

  μ

  ( S )

  = 1 ,   μ

  ( 0 )

  = 0

  where Ø is the subset that contains no element of S (i.e., Ø is the empty set). Note that S corresponds to the “sure” event and 0 to the “impossible” event.

  3.  If Bn ⊂ S, n = 1,2,3,…, k are disjoint, then

  μ

  (

  ∪

  n = 1

  k

  B n

  )

  =

  ∑

  n = 1

  k

  μ

  (

  B n

  )

  (cf. the additive law of probability).

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