Modes at 1 and 0 indicate a non-trivial probability of one team winning or losing every event, which is more common when only a few events occur.
It is trivial that the probability of this state is not zero.
In general, when present, these produce trivial increases in probability of fracture; few produced a precise estimate beyond the pre-test prevalence.
Many authors have paid attention to the subject of stability in probability of trivial solutions of stochastic systems in the past years.
It is easy — all too easy — to assign equal probabilities to the points in such a space: each gets probability 0. Non-trivial probabilities arise when uncountably many of the points are clumped together in larger sets.
Since the order of nonlinearity of (2.2) is more than 1, then obtained stability conditions at the same time are [47 49] conditions for stability in the probability of the trivial solution of nonlinear equation (2.2) and therefore for stability in probability of the equilibrium point of (2.1).
In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger.
The necessary and sufficient conditions for the asymptotic stability in probability of the trivial solution of the quasi-non-integrable-Hamiltonian system are then obtained approximately by examining the sample behaviors of the one-dimensional diffusion process of the square-root of averaged Hamiltonian at the two boundaries.
Since classical statistics employs non-trivial probabilities that attach to the single case in its procedures, a fully frequentists understanding of statistics is arguably in need of a response to the reference class problem.
Individuals assigned to two sources with non-trivial probabilities were considered putative hybrids.
Using the kinetic model of non-trivial surface-polymerization, probabilities of reactions of chemisorbed intermediates are calculated as function of time and carbon number of species, at several sets of reaction parameters and different catalyst properties.
