Rules of Probability

Probability can be defined as a limiting case of a frequency ratio, and from this view the various rules of probability can be derived. An alternative approach is an axiomatic one that states that there is a quantity called probability associated with events and that it possesses assigned properties. The former is largely the frequentist point of view, the axiomatic approach is shared by Bayesians and non-Bayesians alike. 

Probability values lie continuously in the range 0 to 1 inclusive, where the endpoints zero and unity are identified with impossibility and certainty, respectively. This follows immediately for the frequentist for the axiomatic approach it is adopted as an axiom, but one imbued with Laplace s commonsense. Any other range could be chosen at the cost of greater difficulty of interpretation. 

FIGURE 5.2 Venn diagram illustrating the development of conditional probability. 

For the Bayesian, the relationship is taken as an axiom, but its motivation reflects the real world with the foreshadowing of rules implied by the above frequentist treatment. Given the 2 events or propositions, A and B, then 

Application of Uncertainty Analysis to Ecological Risk of Pesticides 

The Rules of Probability Are Recipes for Drawing Consistent Inferences 

I he addition and multiplication rules permit you to calculate the probabilities of certain combinations of events. 

The addition rule holds only if two criteria are met the outcomes are mutually exclu.sive, and we seek the probability of one outcome OR another outcome. 

When they are not di aded by N, the broader term for the quantities n, (/ = 4,B. ) is. statistical weights. If outcomes. A,B.E are both collectively exhaustiv e and mutually exclusive, then 

The multiplication rule applies when the outcomes are independent and we seek the probability of one outcome and another outcome and possibly other outcomes. A more general multiplication rule, described on page 7, applies even w hen outcomes are not independent. 

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At the same time, I would not bet 3 on no rain in return for 4 if it does not rain. This behavior would be inconsistent, since if I did both simultaneously I would bet 6 for a certain return of only 4. Consistent betting would lead me to bet 1 on no rain in remrn for 4. It can be shown that for consistent betting behavior, only certain rules of probability are allowed, as follows. 

There are two central rules of probability theory on which Bayesian inference is based [30] ... 

The process described above is thus repeated with constant time intervals. So, we have a discrete time t = nAr where n is the number of displacement steps. By the rules of probability balance and by the prescriptions of the Markov chain theory, the probability that shows a particle in position i after n motion steps and having a k-type motion is written as follows ... 

Observe Figure 1.2. This figure definitely shows some form of pattern, but is not of such a character that meaningful values can be obtained directly for design purposes. If enough data of this pattern is available, however, they may be subjected to a statistical analysis to predict design values, or probability distribution analysis, which uses the tools of probability. Only two rules of probability apply to our present problem the addition rule and the multiplication rule. 

Addition rule of probability. Now, what is the probability that one event or the other will occur The answer is best illustrated with the help of the Venn diagram, an example of which is shown in Figure 1.4, for the events A and B. There is D, which contains events from A and B it is called the intersection of A and B, designated as A C B. This intersection means that D has events or results coming from both A and B. C has all its events coming from A, while E has all its events coming from B. 

Equation (1.6) is the multiplication rule of probability. If the reduced space is referred to A, then the intersection probability would be... 

Using the intersection probabilities, the addition rule of probability becomes... 

One of the values that is often determined is the value equaled or exceeded. The probability of a value equaled or exceeded may be calculated by the application of the addition rule of probability. The phrase equaled or exceeded denotes an element equaling a value and elements exceeding the value. Therefore, the probability that a value is equaled or exceeded is by the addition rule,... 

The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability that several uncertainties must be combined using the rules of probability and that the calculus of probabilities is adequate to handle all situations involving uncertainty. Probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate... that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability, can better be done with probability. 

The rule that you intuitively arrived at was that if you observed as few as 0 or 1 or as many as 9 or 10 heads out of 10 coin flips, you would conclude that the coin was not fair. How likely is it that such a result would happen In other words, suppose you repeated this experiment a number of times with a truly fair coin. What proportion of experiments conducted in the same manner would result in an erroneous conclusion on your part because you followed the evidence in this way This is the point where the rules of probability come into play. You can find the probability of making the wrong conclusion (calling the fair coin biased) by... 

Using the multiplication rule of probability, together with the above identity, (5) becomes... 

The theorem itself is uncontroversial and follows directly from the multiplication rule of probability but its application to providing probability statements about hypotheses has two difficulties. Suppose we have a particular hypothesis //, in mind and have obtained some evidence E. If we wish to use the theorem to obtain P (//, E) then we must substitute // for A and E for B in the equation of the theorem as given above. The first difficulty is that this requires us to evaluate a so called prior probability of the hypothesis P (// ). The second difficulty is even more severe. We have to evaluate P (E),... 

Mutually exclusive events. Events which stand in that relationship to each other whereby one or other may occur separately but both cannot occur together. For such events the addition rule of probability is simpler than otherwise. 

Yuen, K.-V. and Katafygiotis, L. S. An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability. Probabilistic Engineering Mechanics 20 ) (2005), 109-114. 

We would never have enough data to count all these occurrences, so we make an assumption based on the chain rule of probability (see Appendix A). This states that a sequence as above can be approximated by considered only a fixed window of tags before the current one ... 

Fault trees may be solved by simply applying the rules of probability at each gate. While this method may appear simple, it is important that the unions be carefully considered so that multiple instances of a given probability are calculated only once. An example of this was presented in... 

It is obvious, that radius R distribution makes all R dependent quantities to be spatially inhomogeneous. According to general rules of probability theory [94], the distribution function of some single-valued function Q R) can be expressed via distribution function of R as follows ... 

The Equation 31 means the statistical independence of the first integrals. We remind the reader that two or more events are statistically independent if each individual event is not influenced by the occurrence of any other and Equation 31 corresponds to the well-known rule of probabilities multiplying. 

The probability of an event must be between 0 and 1.00. A probability of zero means an event will not happen. A probability of 1 means an event is certain to happen. (This rule is called the rule of probability range.)... 

At this stage of the simulation, we recognize that the quiescence has been disturbed and the next step is to identify the event responsible for it. The identification of the disturbing event is made rather simply by using the rules of probability theory, viz.,... 

Probabilities can be computed for different combinations of events. Consider one roll of a six-sided die, for example (die, unfortunately, is the singular of dice). The probability that a 4 appears face up is 1/6 because there are N = 6 possible outcomes and only 114 = 1 of them is a 4. But suppose you roll a six-sided die three times. You may ask for the probability that you w ill observ e the sequence of two 3 s followed by one 4. Or >ou may ask for the probabilit> of rolling two 2 s and one 6 in any order. The rules of probability and combinatorics provide the machinery for calculating such probabilities. Here we define the relationships among events that we need to formulate the rules. 

Add Pa, pb and pc pH first OR 4 second) = 5/36-1-5/36-1-1/36 = 1 1/36. This example shows how elementary events can be grouped together into composite events so as to take advantage of the addition and multiplication rules. Reformulation is powerful because virtually any question can be framed in terms of combinations of and and or operations. With these two rules of probability, y ou can draw inferences about a wide range of probabilistic events. 

How should you predict the pij s if you know only the row and column sums The rules of probability tell you exactly what to do. The joint probability Pi, represents the intersection of the set of number j and the set of color i (see Figure 6.4). Each pij is the product of u, the fraction of all possible outcomes that has the right color, and Vj, the fraction that has the right number (see Equation (1.6)),... 

The entropy Sipii,..., pij,... is a function of a set of probabilities. The distribution of p,j s that cause 5 to be maximal is the distribution that most fairly apportions the constrained scores between the individual outcomes. That is, the probability distribution is flat if there are no constraints, and follows the multiplication rule of probability theory if there are independent constraints. If there is a constraint, such as the average score on die rolls, and if it is not equal to the value expected from a uniform distribution, then maximum entropy predicts an exponential distribution of the probabilities. In Chapter 10, this exponential function will define the Boltzmann distribution law. With this law you can predict thermodynamic and physical properties of atoms and molecules, and their averages and fluctuations. How-ever, first we need the machinery of thermodynamics, the subject of the next three chapters. 

Binding Polynomials Can Be Constructed by Using the Addition and Multiplication Rules of Probability... 

You can also get the binding polynomial in Equation (28.15) by using the addition rule of probabilities described in Chapter 1. According to the addition rule, if two states are mutually exclusive (bound and unbound, for example), then y ou can sum their statistical weights, in the same way that terms are summed in partition functions. Use 1 as the statistical weight for the empty site and Kx as the statistical weight for the filled site, and add them to get Q. 

You should follow aU the rules of probability theory in manipulating the data. Do not forget that you can use expert judgment and apply Bayesian updating to give you more accurate numbers. 

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