Probability calculations

The probability calculated so far is too low because it describes one specific sequence of heads and tails. From the point of view of net displacement, the sequence does not matter. Hence the above results must be multiplied by the number of different ways this outcome can arise. Instead of tossing one coin n times, we could toss n coins drawn at random from a piggy bank. For the first, we have a choice of n to draw from for the second, n - 1 for the third, n - 2, and so on. The total possible ways the toss could be carried out is given by the product of these different choices, that is by n ... 

Example 5 Probability Calculation If a six-sided die marked with the mimhers 1, 2, 3, 4, 5, and 6 is thrown, the prohahihty that any given mimher will he uppermost is F6. If the die is thrown twice in succession, then the proh-ahility of a given sequence of mimhers occurring, say, 5 followed hy 6, is (F6)(l/6) = 1/36. The chance of any particular mimher occurring 0, 1, 2, 3, or 4 times in four throws of the die (or in a simultaneous throw of four dice) is given hy the successive terms of Eq. (9-77), expanded as... 

ERROR REDUCTION STRATEGIES. If the error probability calculated by the above procedures leads to an unacceptable overall system failure probability, then the analyst will reexamine the event trees to determine if any PIFs can be modified or task structures changed to reduce the error probabilities to an acceptable level. 

Effects of Improvements in Procedures on Error Probabilities Calculated Using SLIM... 

To illustrate probability calculations involving tlie exponential and Weibull distributions introduced in conjunction willi llie batlitub curve of failure rate, consider first llie case of a mansistor having a constant rate of failure of 0.01 per tliousand hours. To find the probability tliat llie transistor will operate for at least 25,000 hours, substitute tlie failure rate... 

As anollier example of probability calculations - lliis time involving the Weibull distribution - consider a component whose time to failure T in hours lias a Weibull pdf with parameters a = 0.01 and p = 0.50. To find llie probability that llie component will operate for at least 8100 hours, substitute a = 0,01 and p = 0.50 in Eq. (20.4.3), Tliis gives... 

Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

Figure 5. Reaction probabilities for a given instance of the noise as a function of the total integration time Tint for different values of the anharmonic coupling constant k. The solid lines represent the forward and backward reaction probabilities calculated using the moving dividing surface and the dashed lines correspond to the results obtained from the standard fixed dividing surface. In the top panel the dotted lines display the analytic estimates provided by Eq. (52). The results were obtained from 15,000 barrier ensemble trajectories subject to the same noise sequence evolved on the reactive potential (48) with barrier frequency to, = 0.75, transverse frequency co-y = 1.5, a damping constant y = 0.2, and temperature k%T = 1. (From Ref. 39.)...

Table 1. t-test results for GKT with a = 0.05, the numbers shown are the results of 1 — P where P is the probability calculated at the given a. 2002 IEEE. 

The tree for this system was designated Improved Inhibitor System—No Challenges (Figure 8). Probability calculated from this fault tree indicated that fewer than 3 out of 10,000,000 cars would fail with this system. Again, these changes were relatively simple and cost very litde to implement. I should mention again that we do not claim that these probabilities are accurate, but because the fault trees and the input data are consistent the comparisons remain meaningful and the results are dramatic. 

Our data were 15 heads and 5 tails, so how do we calculate the p-value Well, remember the earlier definition and translate that into the current setting the probability of getting the observed data or more extreme data in either direction with a fair coin. To get the p-value we add up the probabilities (calculated when the null hypothesis is true - coin fair) associated with our data (15 heads, 5 tails) and more extreme data (a bigger dilference between the number of heads and the number of tails) in either direction ... 

It is useful to look at this visually. Figure 3.2 plots each of the outcomes on the x-axis with the corresponding probabilities, calculated when the null hypothesis is true, on the y-axis. Note that the x-axis has been labelled according to heads -tails (H —L), the number of heads minus the number of tails. This identifies each outcome uniquely and allows us to express each data value as a difference. More generally we will label this the test statistic, it is the statistic that the p-value calculation is based. The graph and the associated table of probabilities are labelled the null distribution (of the test statistic). 

The upper bound of the escape probability, corresponding to t oo, can also be analytically calculated. In this limit, the electron is allowed to make many revolutions in the same orbit around the cation before it is scattered into another orbit. Under this condition, the electron motion may be described as diffusion in energy space [23]. The escape probability calculated by using the energy diffusion model is also included in Fig. 3. We see that the simulation results for finite x properly approach the energy diffusion limit. 

It must be stressed, because this error is often made, that the probability calculated above is not the probability of the truth of Hq. It is the probability that, given the truth of Hp, in a repeated measurement an equal or more extreme value of the test statistic will be found. 

There are three ways to perform a t test in Excel. First, a t value can be calculated from the appropriate equation and then the probability calculated from =TDIST (t, df, tails ), where df = degrees of freedom and tails = 2 or... 

Note that p(xi rj) denotes the posterior probability calculated using Bayes rule and the above equations clearly convey the centroid aspect of the solution. 

The widening of the energy levels corresponding to the transition probabilities calculated from this expression showed satisfactory agreement with experiment. This suggests that the proposed mechanism is probably correct. 

This corresponds to a one-dimensional random walk with an absorbing barrier, with the transition probabilities calculated from quantum mechanics. The time dependent distribution of the reactant molecules among... 

Normally one might expect that if the transition probability vanishes on resonance it also vanishes off resonance. However, such is not the case. When the transition probability is calculated off resonance, by numerically solving Eqs. (14.16) using a Taylor expansion method, it is nonzero for both v E and v 1E.14,16 In Fig. 14.6 we show the transition probabilities obtained using two different approximations for v E, and vlE for the 17s (0,0) collisional resonance.16 To allow direct comparison to the analytic form of Eq. (14.21) we show the transition probabilities calculated with EAA = VBB = 0. For these calculations the parameters ju2l = pLz, = 156.4 ea0, b = 104ao, and v = 1.6 x 10-4 au have been used. The resulting transition probability curves are shown by the broken lines of Fig. 14.6. As shown by Fig. 14.6 these curves are symmetric about the resonance position. The vlE curve of Fig. 14.6(b) has an approximately Lorentzian form, but the v E curve of Fig. 14.6(a), while it vanishes on resonance as predicted by Eq. (14.24), has an unusual double peaked structure. 

The observed cross sections for the 18s (0,0) collisional resonance with v E and v 1 E are shown in Fig. 14.12. The approximately Lorentzian shape for v 1 E and the double peaked shape for v E are quite evident. Given the existence of two experimental effects, field inhomogeneties and collision velocities not parallel to the field, both of which obscure the predicted zero in the v E cross section, the observation of a clear dip in the center of the observed v E cross section supports the theoretical description of intracollisional interference given earlier. It is also interesting to note that the observed v E cross section of Fig. 14.12(a) is clearly asymmetric, in agreement with the transition probability calculated with the permanent electric dipole moments taken into account, as shown by Fig. 14.6. 

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