Definition of Probability

Assume that one repeats an experiment many times and observes whether or not a certain event x is the outcome. The event is a certain observable result defined by the experimenter. If the experiment was performed N times, and n results were of type x, the probability P(x) that any single event will be of type x is equal to 

The ratio n /N is sometimes called the relative frequency of occurrence of x in the first N trials. 

There is an obvious difficulty with the definition given by Eq. 2.1—the requirement of an infinite number of trials. Clearly, it is impossible to perform an infinite number of experiments. Instead, the experiment is repeated N times, and if the event x occurs n times out of N, the probability P(x) is 

Equation 2.2 will not make a mathematician happy, but it is extensively used in practice because it is in accord with the idea behind Eq. 2.1 and gives useful results. 

As an illustration of the use of Eq. 2.2, consider the experiment of tossing a coin 100 times and recording how many times the result is heads and how many it is tails. Assume that the result is 

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Misesian definition of probability that equation 2.4-6 will result for this case as may be demonstrated by adding two ncai-disjoint probabilities (refer to the Veni diagram, Figure 2.4-1). The segments are P(A,) = P(A) P(B), P(Af) = P(A) P B), and... 

Thus, the conversion from a logical representation to a probability representation follows naturally from superposition and the von Misesian definition of probability. Simply replace tlie component identifier by its failure probability and combine probabilities according to the logical operations. 

Thus, the sum of the probabilities P, equals unity as it must from the definition of probability. For a continuous set of eigenkets, this relationship is replaced by... 

What is the probabiUty of finding a specific site, say the ith site, occupied The answer can be given by using the so-called classical definition of probability... 

Next, we seek the probability of finding two specific sites i and j simultaneously occupied. This can be calculated again by the classical definition of probability. 

There exists a much simpler example. Let x and y be independent positive real number. This means that vector (x, y) is uniformly distributed in the first quadrant. What is probability that x y Following the definition of probability based on the density of sets, we take the correspondent angle and find immediately that this probability is Vi. This meets our intuition well. But let us take the first number x and look for possible values of y. The result for given x the... 

Throughout this book, the approach taken to hypothesis testing and statistical analysis has been a frequentist approach. The name frequentist reflects its derivation from the definition of probability in terms of frequencies of outcomes. While this approach is likely the majority approach at this time, it should be noted here that it is not the only approach. One alternative method of statistical inference is the Bayesian approach, named for Thomas Bayes work in the area of probability. 

From the definition of probability the expected number of trial function evaluations, T, required to encounter an X +, such that A4> < 0 equals / . To estimate the growth of T with n it is convenient to represent it in the following form ... 

A simple but effective definition of probability is based on a frequency of occurrence concept. If an event A can occur in cases out of N possible and equally probable number of cases, the probability P A) that the event will ocx ur in any new trial is... 

There has been much controversy about the proper definition of probability. One definition is the following If an experiment has n equally probable outcomes, m of which are favorable to the occurrence of a certain event A, then the probability that A occurs is m/n. Note that this definition is circular, since it specifies equally probable outcomes when probability is what we are attempting to define. It is simply assumed that we can recognize equally probable outcomes. An alternative definition is based on actually performing the experiment many times. Suppose that we perform the experiment N times and that in M of these trials the event A occurs. The probability of A occurring is then defined as... 

Realizations of the potentials of hazards have various occurrence probabilities, and severities of consequences. Definitions of probability and severity can be tailored to suit particular needs. 

The values become more certain as the number of trials is increased. A definition of probability based on this concept is stated in Equation B-2 ... 

The definition of probability as relative frequency implies that the event under investigation—here the failure of a component—must have occurred several times lest the confidence intervals be too large. If a component has rarely failed an evaluation using Bayes theorem is appropriate [25]. It is based on the so-called subjective notion of probability. 

The frameworks can provide useful guidance, but their scientific basis deserves further work. If a concept is introduced in a framework it should be properly defined. The frameworks are for instance unclear about definitions of probabilities and uncertainties. Referring to a probability is not sufficient as probabilities can be interpreted in different ways. And depending on the chosen interpretation, we are led in different directions for assessing risk. 

Definition of probability matrix of initial consequence states is needed ... 

Concept of Probability and Rules of Computation. The definition of probability is determined as application-oriented. Logic difficulties resulting as compared to exact mathematical definition on quantity-theoretical basis, can be ignored in the present context. If A is the designation of an event as the result of an experiment with clearly described objects under... 

The classical definition of probability states that the probability P(A) of an event A is determined a priori without actual experimentation. It is given by... 

The axiomatic definition of probability uses the set theory. A certain event X is the event that occurs in every trial. The union A + B of two events A and B is the event that occnrs when A or B both occur. The intersection AB of the events A and B is the event that occurs when both events A and B occur. The events A and B are mutually exclusive if the occurrence of one of them excludes the occurrence of the other. Three postulates are given. The probability P(A) of an event A is... 

Frankly speaking, Bayes rule involves the manipulation of conditional probabilities. However, there exists a sharp difference in classical and Bayesian approaches. The major difference between the Bayesian and classical approaches to statistical inference is in the definition of probability. The classical approach asserts that the probability of a fair coin tossed, landing heads is 14 after repetitive tests. In contrast to this, as per Bayesian approach, will say that the probability a coin lands heads is Vz expressing a degree of belief, and would argue that based on the symmetry of the coin, there is no reason to think that one side is more likely to come up than the other side. This definition of probability is usually termed subjective probability [1]. The classical approach uses probability to express the frequency of certain types of data to occur over repeated trials. The Bayesian approach uses probability to express belief in a statement about unknown quantities. This will be clearer from the difference between the two approaches which have divided statistical approaches in two clear divisions, i.e. now in statistics there are two schools of thoughts ... 

From the definition of probability distributions, it is evident that ... 

There is no universally accepted definition of probable maximum loss (PML) for purposes of earthquake risk analysis, but it is often understood to mean the loss with 90 % nonexceedance probability given shaking with 10 % exceedance probability in 50 years. For a single asset, PML can be calculated from the seismic vulnerability function by inverting the conditional distribution... 

The nature and amount of stractural damage depends necessarily on the quality of the materials that compose stractural and nonstractural elements, and on the configuration and type of stractural systems. In the early years of modem earthquake engineering, damage definition was basically approached in qualitative terms (e.g., through the definition of probable localization of such damage in a structure). This kind of approach relied fundamentally on the observation of damaged stmctuies after seismic events or it... 

If the probability of some event A is nJN and the probability of event B is n /N, the probability that A and B will occur is the product of the probability of A and the probability of B, that is, (n /N) x nJN). This also follows from the definition of probability. Consider that there are four ways to categorize all possible events (1) A and B occur, (2) A but not B occurs, (3) B but not A occurs, and (4) neither A nor B occurs. The probability that B will not occur is 1 less the probability that B will occur (1 - fig/N), and likewise that A will not occur. With designating the probability of the fth category events, several conditions exist because of the definition of the probability. 

Let us now consider many sources of electrons uniformly distributed at all distances from the surface of the solid and detect those unscattered electrons which emerge normal to the surface. What sort of distribution of the depths of these electrons can be detected This new function, P d), will have the same exponential form as P d) since the detection of electrons from different depths in the solid is directly proportional to the probability of electron escape from each depth. What percentage of electrons will have come from within a distance of one IMFP from the surface Recall from our definition of probability that this is simply the integral between the limits of 0 and 1 in the exponential function divided by the integral over all space... 

Let us relate this definition of probability to a hypothetical series of measurements. Suppose that we measure on a beam balance the mass of a block of aluminum and that we perform this measurement over and over for a total of 50 measurements. In this example, we are going to assume that the actual mass of the metal does not change (i.e., pieces do not break off, nor does the block get dirty from handling) assume also, to illustrate a point, that the measurements spread over a larger range than we would most probably see if we used a good laboratory balance. The results of the 50 measurements are listed in Table 12-1. These data also are plotted as bar graphs in Fig. 12-1. 

There are experiments with more than one outcome for any trial. If we do not know which outcome will result in a given trial, we define outcomes as random and we assign a number to each outcome, called the probability. We present two distinct definitions of probability ... 

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