Multiplication rules of probability

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 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),... 

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... 

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. 

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... 

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. 

There are different restriction enzymes that cut DNA at different sites. The previous sequence can be repeated several times for the same DNA sample. From a study of each restriction enzyme, a probability that another person will have the same profile is assigned. Thus, one restriction enzyme may have the possibility that another person has the same match of 1 in 100 or 1%. A second restriction enzyme may have the probability of 1 in 1000 or 0.1%. A third restriction enzyme may have a probability for a match of 1 in 500 or 0.2%. If there is a match with all three restriction enzymes, the probability would be 0.01 x 0.001 x 0.002 or 0.00000002% or 0.000002% or 1 part in 50,000,000. There is a caution to using the multiplication rule, in that DNA sequences are not totally random. In fact, DNA sequence agreements generally diverge as one s ancestors are less closely related. 

As a general rule, the voucher agency should receive a copy of all forms. Service providers will probably wish to keep a copy of the form for their records. Sometimes diagnostic service providers will require copies of at least parts of the clinic record. Multiple copies of forms can be produced cheaply using NCR (no carbon required) paper that prints through to the sheet below with the pressure of a pen. 

The two latter forms of Bayes rule provide an analytically simple way of combining multiple sources of evidence. Bayesian inference becomes much more difficult when the evidence is not certain or when the conditional independence assumption is not met. When evidence is not certain, complex multistage forms of Bayesian analysis are required that consider the probability of the evidence being true (Winterfeldt and Edwards 1986). When conditional independence is not true, the expanded form of Bayes rule must be modified. For example, consider the case where the evidence consists of three events ( , E2, 3), where E, and E are conditionally dependent and E, is conditionally independent of the two other events. The posterior probability, E(//j ,E2 3)> then becomes ... 

Equation (15) can be extended to multiple operators or multiple sources of evidence (Lehto and Papastavrou 1991). The resulting expression takes into account the probability of a felse edeirm and the probability of detection for the other source of information. Lehto and Papastavrou use this approach to analyze situations where the other source of information is a warning signal. The extent to which human judgments correspond to the predictions of Bayes rule is further discussed in Section 4.1. 

One of the selection rules that governs the transition probability is that the spin of the electrons must be retained, i.e. the electron spins in the grotmd state -n bonding orbital are paired and antiparallel and they retain their antiparallel spin in the transition to the excited tt-tt state. The multiplicity, M, of these states describes the spin states of the electrons. Electrons are designated to have a spin quantum number, S, of either +V2 or -V2 and so the multiplicity of the state can be calculated according to equation 1 ... 

There are two very simple rules or laws used in the calculation of probability, known respectively as the law of addition and the law of multiplication. 

From this expression it is easy to derive the multiplication rule for independent events. Suppose A is independent of B. Then information about B gives no information about A. That is, the conditional probability of A given B is the same as the probability of A or (by formula)... 

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

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. 

Some problems in probability cannot be solved directly by applying the addition or multiplication rules. Such questions ctm usually be reformulated in terms of composite events to which the rules of prohability can be applied. Example 1.7 shows how to do this. Then on page 13 w e ll use reformulation to construct prohability distribution functions. 

EXAMPLE 1.7 Elementary and composite events. What is the probability of a 1 on the first roll of a die or a 4 on the second roll If this were an and question, the probability would be (1/6)(1/6) = 1 /36, since the two rolls are independent, hut the question is of the or ty pe, so it cannot he answ ered by direct appheation of either the addition or multiplication rules. But by redefining the problem in terms of composite events, you can use those rules. An individual coin toss, a single die roll, etc. could be called an elementary event. A composite event is just some set of elementary events, collected together in a convenient way. In this example it s convenient to define each composite event to be a pair of first and second rolls of the die. The advantage is that the complete list of composite events is mutually exclusive. That allows us to frame the problem in terms of an or question and use the multipUcation and addition rules. The composite events are ... 

General Multiplication Rule (Bayes Rule). If outcomes A and B occur with probabilities piA) and p(B), the Joint probability of events 4 and B is... 

If ev ents. 4 and B happen to be independent, the pre-condition A has no influence on the probability of B. Then p(B. 4) = p(B), and Equation (1.11) reduces to piAB) = p(B)piA), the multiplication rule for independent events., 4 probability p B) that is not conditional is called an a priori probability. The conditional quantity p(B A) is called an a posteriori probability. The general multiplication rule is general because independence is not required. It defines the probability of the intersection of events, piAB) = p A n B). 

The second equality in Equation (1.12) follows from the general multiplication rule, Equation (1.11). i g = 1, ev ents A and B are independent and not correlated. If > 1, events A and B are positively correlated. If < 1, events A and B are negatively correlated. HA = 0 and A occurs then B will not. If the a priori probability of rain is piB) = 0.1, and if the conditional probcibility of rain, given that there are dark clouds. A, is p(B A) = 0.5, then the degree of correlation of rain with dark clouds is = 5. Correlations are important in statistical thermodynamics. For example, attractions and repulsions among molecules in liquids can cause correlations among their positions and orientations. 

Probabilities describe incomplete knowledge. The addition and multiplication rules allow you to draw consistent inferences about probabilities of multiple e ents. Distribution functions describe collections of probabilities. Such functions have mean values and variances. Combined with combinatorics—the counting of arrangements of systems—probabilities provide the basis for reasoning about entropy, and about driving forces among molecules, described in the next chapter. 

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