Coincidence probability of streamflow in water resources area, water receiving area and impacted area: implications for water supply risk and potential impact of water transfer

Under changing environment, the feasibility and potential impact of an inter-basin water transfer project can be evaluated by employing the coincidence probability of runoff in water sources area (WSA), water receiving area (WRA), and the downstream impacted area (DIA). Using the Han River to Wei River Water Transfer Project (HWWTP) in China as an example, this paper computed the coincidence probability and conditional probability of runoff in WSA, WRA and DIA with the copula-based multivariate joint distribution and quantified their acceptable and unfavorable encounter probabilities for evaluating the water supply risk of the water transfer project and exploring its potential impact on DIA. Results demonstrated that the most adverse encounter probability (dry–dry–dry) was 26.09%, illustrating that this adverse situation could appear about every 4 years. The acceptable and unfavorable probabilities in all encounters were 44.83 and 55.17%, respectively, that is the unfavorable situation would be dominant, implying flood and drought risk management should be paid greater attention in project operation. The conditional coincidence probability (dry WRA & dry DIA if dry WSA) was close to 70%, indicating a requirement for an emergency plan and management to deal with potential drought risk.

On the probability of coincidence of cycle lengths for independent random permutations with given number of cycles

From the set of all permutations of the degree n with a given number N ≤ n of cycles two permutations are choosed randomly, uniformly and independently. The cycles of each permutation are numbered in some of N! possible ways. We study the coincidence probability of the cycle lengths of permutations for a given numbering. This probability up to a suitably selected renumbering of cycles of the first permutation equals to the probability of similarity of these permutations. The asymptotic estimates of the coincidence probability of the cycle lengths are obtained for five types of relations between N, n → ∞.

GENERALIZED BIRTHDAY PROBLEMS IN THE LARGE-DEVIATIONS REGIME

This paper considers generalized birthday problems, in which there aredclasses of possible outcomes. A fractionfiof theNpossible outcomes has probability αi/N, where$\sum_{i=1}^{d} f_{i} =\sum_{i=1}^{d} f_{i}\alpha_{i}=1$. Samplingktimes (with replacements), the objective is to determine (or approximate) the probability that all outcomes are different, the so-calleduniqueness probability(or:no-coincidence probability). Although it is trivial to explicitly characterize this probability for the cased=1, the situation with multiple classes is substantially harder to analyze.Parameterizingk≡aN, it turns out that the uniqueness probability decays essentially exponentially inN, where the associated decay rate ζ follows from a variational problem. Only for smalldthis can be solved in closed form. Assuming αiis of the form 1+φiɛ, the decay rate ζ can be written as a power series in ɛ; we demonstrate how to compute the corresponding coefficients explicitly. Also, a logarithmically efficient simulation procedure is proposed. The paper concludes with a series of numerical experiments, showing that (i) the proposed simulation approach is fast and accurate, (ii) assuming all outcomes equally likely would lead to estimates for the uniqueness probability that can be orders of magnitude off, and (iii) the power-series based approximations work remarkably well.