Expectation probability curve

Figure 6.7 Expectation-probability curve for decision-making to adopt the means to improve.

It is clarified from Figure 6.3 that when the installation cost is half the profit after a successful installation (A = B/2), decision-making in the region of the probability of success of 0.4 < P < 0.6 is not easy this is because the slope of the expectation-probability curve is quite small in this region. On the other hand, it is easy to make a decision in the region of the probability of success of P < 0.4 or P > 0.6 because the slope of the expectation-probability curve is clearly large in this region. 

Figure 6.9 Expectation-probability curve of betting for certain condition that gives reasonable explanation.

Expectation curves are alternatively known as probability curves . This text will use the term expectation curve for conciseness. 

Figure 1.22. The null and the alternate hypotheses Hq resp. Hi. The normal distribution probability curves show the expected spread of results. Since the alternate distribution ND(/tb, a might be shifted toward higher or lower values, two alternative hypotheses Hi and H are given. Compare with program HYPOTHESIS. Measurement B is clearly larger than A, whereas S is just inside the lower CL(A).

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. 

Curve A is clearly the solubility curve of C-S-H(I) scatter at the CaO-rich end may arise because a given Ca/Si ratio can be associated with more than one structural arrangement. However, any form of C-S-H that is structurally derived from 1.4-nm tobermorite or jennite, or perhaps more widely that is based on Ca-0 sheets, probably has a solubility lying on or close to this curve. There are no solubility data for C-S-H(II), 1.4-nm tobermorite or jennite, but one might expect the curves for the crystalline... 

Fig. 14.4), it is best shown by the radii of the -1-3 cations (Fig. 14.5). There are two noticeable dilTerenccs between the two series of ions (1) although the actinide contraction initially parallels that of the lanthanides, the elements from curium on are smaller than might be expected, probably resulting from poorer shielding by 5/ electrons in these elements (2) the lanthanide curve consists of two very shallow arcs with a discontinuity at the spherically symmetrical Gd " (4/ ) ion. A similar discontinuity is not clearly. seen at... 

The run-down method was developed at Frankford Arsenal, Philadelphia, and produces a more complete frequency and probability of reaction curve [22]. Although a larger number of tests is used than in the up-and-down procedure, the method makes possible a better evaluation of the distribution of the population. Starting at a level expected to be between 0% and 100% reactions levels, 20-25 tests are made at each of several levels above and below the starting level until the 0% and 100% levels are reached. The increments between the levels normally are equal to or less than the expected standard deviation. A cumulative probability curve is then plotted from the results of the entire test which is considered to be the frequency distribution of the population. 

Keywords deterministic methods, STOllP, GllP, reserves, ultimate recovery, net oil sands, area-depth and area-thickness methods, gross rock volume, expectation curves, probability of excedence curves, uncertainty, probability of success, annual reporting requirements, Monte-Carlo simulation, parametric method... 

It is common practice within oil companies to use expectation curves to express ranges of uncertainty. The relationship between probability density functions and expectation curves is a simple one. 

Figure 6.6 The probability density function and the expectation curve...

An alternative and commonly used representation of the range of reserves is the proven, proven plus probable, and proven plus probable plus possible definition. The exact cumulative probability which these definitions correspond to on the expectation curve... 

When an explorationist constructs an expectation curve, the above approach for the volumetries of an accumulation is taken, but one important additional parameter must be taken into account the probability of there being hydrocarbons present at all. This probability is termed the Probability of Success" (POS), and is estimated by multiplying together the probability of there being ... 

The estimated probabilities of each of these events occurring are multiplied together to estimate the POS, since they must a//occur simultaneously if a hydrocarbon accumulation is to be formed. If the POS is estimated at say 30%, then the probability of failure must be 70%, and the expectation curve for an exploration prospect may look as shown in figure 6.9. 

As for the expectation curve for discoveries, the success part of the probability axis can be divided into three equal sections, and the average reserves for each section calculated to provide a low, medium and high estimate of reserves, if there are hydrocarbons present. 

The same questions may then be asked for different values of the probabilities p and po. The answers to these questions can give an indication of the importance to the company of P at various levels of risk and are used to plot the utility curve in Fig. 9-25. Positive values are the amounts of money that the company would accept in order to forgo participation. Negative values are the amounts the company woiild pay in order to avoid participation. Only when the utihty value and the expected value (i.e., the straight line in Fig. 9-25) are the same can net present value (NPV) and discounted-cash-flow rate of return (DCFRR) be justified as investment criteria. 

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