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By the average absolute difference between the unconditional failure probability and the conditional one on fixing an input at its realization, the failure-probability-based-sensitivity (FP-S) is defined to quantify the effect of the fixed input on the failure probability, which provides important information for reliability-based design optimization of the structure. Among the estimation methods for FP-S, the Bayes theorem-based methods are competitive, but the conditional probability density function (PDF) should be estimated in this type method. To alleviate the computational complexity of estimating conditional PDF, a novel FP-S estimation method is proposed by use of the conditional probability theorem. In the proposed method, the conditional failure probability on fixing the input at its realization is approximated by the conditional failure probability on fixing the input in a small interval, in which the conditional probability theorem of the random event can be used to transform FP-S as estimations of a series of probabilities, and they can be simultaneously completed by a numerical simulation for estimating the unconditional failure probability. For ensuring the precision of the approximation introduced by replacing the realization with the small interval, a selection strategy for the small interval is proposed. Comparing with the competitive Bayes theorem-based estimation for FP-S, the proposed method replaces the conditional PDF estimation with the conditional probability estimation, which greatly reduces the computational complexity and improves the accuracy of the FP-S estimation. By combining with the adaptive kriging surrogate model, the efficiency of the proposed method can be drastically improved, and the presented examples demonstrate the efficiency and accuracy of the proposed method.

This study presents the rainfall statistics, conditional probability structure and statistical dependence of rainfall amount of several gauging stations located around Peninsular Malaysia, namely Subang, Senai and Kota Bharu. Daily rainfall measurements for all stations were collected from the Department of Meteorology, Malaysia are long and reliable, with at least 40 years of data. The average annual rainfall estimated for Kota Bharu, Subang and Senai are 2,627 ± 574 mm, 2581 ± 399 mm and 2499 ± 340 mm, respectively. The effect of monsoon seasons on the monthly rainfall amount is evident in this study. The most significant variation in the average monthly rainfall is noticed for Kota Bharu. There was some variation in the average monthly rainfall for Subang and Senai. The conditional probability structure for t-consecutive wet and dry days show that the multi-day events are time-dependent. For example, the probability of occurrence for a single dry day is 0.458, 0.453 and 0.553 and increased significantly to 0.696, 0.780 and 0.817 for 8-consecutive dry days at Subang, Senai and Kota Bharu, respectively. The dependence of rainfall amount was analyzed using the auto correlation function (ACF). The range of ACFs estimated for all stations were very low, i.e. 0.0050 to 0.0209, 0.0093 to 0.0857 and 0.0633 to 0.3700 for Subang, Senai and Kota Bharu, respectively. This result shows that the rainfall amounts are independent of each other. Overall, the analysis shows that the east coast region received more annual rainfall with higher variability, as compared to the central and south parts of Peninsular Malaysia. Additionally, the total amount of rainfall observed for all stations varies spatially and temporarily.

We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.

In the context of global warming, drought events occur frequently. In order to better understanding the process and mechanism of drought occurrence and evolution, scholars have dedicated much attention on drought propagation, mainly focusing on drought propagation time and propagation probability. However, there are relatively few studies on the sensitivities of drought propagation to seasons and drought levels. Therefore, we took the Heihe River Basin (HRB) of Northwest China as the case study area to quantify the propagation time and propagation probability from meteorological drought to agricultural drought during the period of 1981–2020, and subsequently explore their sensitivities to seasons (irrigation and non-irrigation seasons) and drought levels. The correlation coefficient method and Copula-based interval conditional probability model were employed to determine the drought propagation time and propagation probability. The results determined the average drought propagation time as 8 months in the whole basin, which was reduced by 2 months (i.e., 6 months) on average during the irrigation season and prolonged by 2 months (i.e., 10 months) during the non-irrigation season. Propagation probability was sensitive to both seasons and drought levels, and the sensitivities had noticeable spatial differences in the whole basin. The propagation probability of agricultural drought at different levels generally increased with the meteorological drought levels for the upstream, midstream, and southern downstream regions of the HRB. Lesser agricultural droughts were more likely to be triggered during the irrigation season, while severer agricultural droughts were occurred mostly during the non-irrigation season. The research results are helpful to understand the characteristics of drought propagation and provide a scientific basis for the prevention and control of droughts. This study is of great significance for the rational planning of local water resources and maintaining good ecological environment in the HRB.

Promoting the greening of industry is the key to achieving high-quality and sustainable development of the urban economy. It is particularly important for resource-based cities (RBCs) that exploit natural resources as the leading industries. In this paper, the Windows-Bootstrap-DEA model was used to calculate the industrial eco-efficiency (IEE) of 114 RBCs in China from 2003 to 2016, and the regional differences and dynamic evolution characteristics of the IEE were analyzed. The panel Tobit model was used to explore the factors associated with IEE in RBCs. The results showed that the IEE of RBCs in China was at a low level during the study period, and the resource utilization process had not reached an optimal state. There were large regional differences in IEE, and there was a significant degree of spatial agglomeration. The results of conditional probability density estimation showed that the distribution of IEE had strong internal stability on the whole, and the distributions of IEE of RBCs in different regions, different resource types, and different development stages showed significant differences. The results of the panel Tobit model showed that per capita GDP, ownership structure, science and technology input, and industrial agglomeration had significant positive effects on IEE, while industrial structure and employment structure showed significant negative effects. The conclusions of this paper can provide a scientific decision-making basis for industrial transformation planning of RBCs.

Odor pollution, also referred to as odor nuisance, is a growing environmental concern that is significantly associated with mental health. Once emitted into the air, the concentration of odorous substances varies considerably with wind conditions, leading to difficulties in timely sampling. In the present study, we employed selected ion flow tube mass spectrometry (SIFT-MS) to measure 22 odor-producing molecules continuously in an urban–rural complex city. In addition, we applied statistical analyses, principal component analysis (PCA), and a conditional probability function (CPF) to the datasets obtained from SIFT-MS to identify the odor characteristics at two study sites. At site A, odorants related to livestock farming and industry showed high factor loadings on principal components (PCs) from the PCA. In contrast, we estimated that the odorous gaseous chemicals affecting site B were closely related to sewage treatment and municipal solid waste disposal. Similar CPF patterns of grouped substances from the PCA supported the association between potential odor sources and specific odorants at site B, which helped estimate possible source locations. Consequently, our findings indicate that continuous monitoring of odorous substances using SIFT-MS can be an effective way to provide sufficient information on odor-producing molecules, leading to the clear identification of odor characteristics despite the high variability of odorous substances.

Multiple representations can enhance students’ understanding of mathematical concepts and complex information but can also pose well-documented challenges for students. Whereas instructional designs have been optimized to support students’ learning with multiple representations, little is known about supportive teaching practices for dealing with multiple representations in whole-class discussions. In this article, we qualitatively investigate two cases of teacher-student interaction in whole-class discussions in grades 10–12 (about the mathematical topic of complex conditional probability information). The analysis aims at decomposing the teaching practices into those actions that can support or hinder students’ understanding. The comparison of cases reveals that teaching practices can vary greatly: simply translating compacted concepts of a given text into other representations (visual area model, symbolic representation of fractions, and three language varieties) seems to be sufficient for students with advanced understanding. Other students need teachers’ supportive actions for unfolding the highly compacted concepts (such as part-of-part) into several concept elements (part, whole, and part-whole relationship) and explicitly connecting (rather than only translating) the concept elements in multiple representations for the different concept elements. The findings can inform both theory building on teaching practices with multiple representations and professional development.

At present, there are few fire risk assessment models for historical buildings with wooden components that are highly vulnerable to fires. Bayesian network (BN) is a practical risk assessment method that is helpful to risk management. Because many indicators affect the fire risk of historical buildings, the BN may have many nodes and a complex structure. As the parent nodes increase, the number of conditional probability distributions required to determine the child node’s conditional probability table (CPT) grows exponentially. Therefore, a combined method is proposed to reduce the complexity of determining CPTs by integrating various methods, including historical data collection, expert knowledge, logical relationships, and empirical formulas. Based on the proposed combined method, combining fire protection codes and provisions in China, a BN-based fire risk assessment model is presented for historical buildings, which integrates static indicators, such as building parameters and cultural significance, and dynamic indicators, such as environmental factors. In the model, the variable weight synthesis theory is introduced to automatically adjust the nodes’ relative importance (weights) to improve the rationality of the BN for the special scenarios and the unconventional emergency scenarios. The proposed BN model can reflect the fire development and spread trend, assess the fire risk of historical buildings, analyze key factors, and propose fire improvement measures. Taking Nanyue Temple as a case study, based on the collected field data, the assessment results have demonstrated and verified the proposed model and method are feasible. The sensitivity analysis results have verified the rationality of the method and identified the key factors.

Water resources in China, especially in the major basins of the Songhua and Yangtze rivers, are characterized by uneven distribution both temporally and spatially, leading to notable challenges in per capita water availability. In this study, we investigate the ensemble probability distribution of annual runoff over the past 70 years in two of China's major watersheds: the Songhua River and the Yangtze River. By dividing each basin into several regions from upstream to downstream, based on annual mean discharge as a proxy for annual runoff, we observed a significant correlation between the annual runoff of the control sections and those of the upstream and downstream regions in these large watersheds. Consequently, this study establishes the probability distribution of annual runoff for each region within the basins, anchored on the design annual runoff of the control sections, using the joint bivariate logarithmic normal distribution of two interrelated random variables. Furthermore, we developed a relationship between the conditional probability distribution and correlation diagram data, and a regression equation correlating regional annual runoff with the watershed control section's runoff. This investigation into the historical patterns of runoff over the past 70 years provides a comprehensive understanding of the dynamics in these critical watersheds.

The conditional probability interpretation of quantum gravity has been criticized for violating the constraints of the theory and also not giving the correct expression for the propagator. We have shown that following Page’s proposal of constructing an appropriate projector for the stationary state of a closed system, we can arrive at the correct expression for the propagator by using conditional probability rule. Also, it is shown that a unitary evolution of states of a subsystem at local level may be a consequence of non-unitary projection of appropriate states at global level.