Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given the first four rolls turn up heads, the probability that the next toss is a head is in fact,

Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

: Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.

Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.

Given a parameterized family of probability density functions ( or probability mass functions in the case of discrete distributions )

Given a probability space and a measurable space,

Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

Solomonoff's universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

Given that the probability p is known, past outcomes provide no information about future outcomes.

Given a confidence level, the VaR of the portfolio at the confidence level is given by the smallest number such that the probability that the loss exceeds is at most.

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information.

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

Given a probability space with, the indicator random variable is defined by if otherwise

Given a discrete-time stationary ergodic stochastic process on the probability space, AEP is an assertion that

Given the observed percentage difference p − q ( 2 % or 0. 02 ) and the standard error of the difference calculated above (. 03 ), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0. 02 and standard deviation 0. 03 is greater than 0.

: Given a random galaxy in a location, the correlation function describes the probability that another galaxy will be found within a given distance.

Given all this information, the probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs ( for a universal computer ) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes ' theorem can be used to predict the yet unseen parts of x in optimal fashion.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given an operator on Hilbert space, consider the orbit of a point under the iterates of.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

Given MLB's refusal to consider expansion, Kirksey, Cullinan, Smith, and Hofheinz joined forces with would-be owners from other cities and announced the formation of a new league to compete with the established National and American Leagues.

Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

Given the well-publicized hoopla, some sources still consider this as a nomination for Davis ; however, the Academy does not officially record this as a nomination.

Given the importance of these concepts in Moral Politics, it is important to consider their meaning along with how each view suggests and is justified by a corresponding view of the nature of child rearing, morality, and justice.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

" Given that abstract thought is thus limited, he went on to consider how historical formations give rise to different philosophies and ways of thinking.

Given Milan's notoriously damp and foggy climate, the Milanese consider it a fair-weather day when the Madonnina is visible from a distance, as it is so often covered by mist.

Given the Protestant belief of justification through faith alone, versus on the basis of merit, most Protestant Christians consider themselves saved without having to keep the commandments of the Mosaic Law as a whole.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given its native distribution the species is also a potential natural vagrant to Western Europe and there have been records in areas such as Cornwall and the Isles of Scilly which some observers consider may relate to wild birds ; however, given the Wood Duck's popularity in captivity it would be extremely difficult to prove their provenance one way or another.

Labor Party News quoted Leonard Riley, President of the Charleston International Longshoremen's Association Local 1422 as saying, " Given the results of the past few elections, I think the workers of South Carolina would jump at the opportunity to consider a Labor Party which would guarantee an uncompromising voice for working people on their issues.

Given that we have been on " yellow alert " for over two and a half years now, we cannot consider this a particularly exceptional condition that warrants curtailment of constitutional rights.

Given a relation of order N, consider a projection that subtends X, Y, and Z as the key and W as the residual attribute.

Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II < sub > 1 </ sub > factor.

Given that the prominent Uranians were trained Classicists, I consider ludicrous the view, widely held, that ‘ Uranian ’ derives from the German apologias and legal appeals written by Karl-Heinrich Ulrichs in the 1860s, though his coinage Urning — employed to denote ‘ a female psyche in a male body ’ — does indeed derive from the same Classical sources, particularly the Symposium.

Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals / flows which agree on shell.

Given a line and any point A on it, we may consider A as decomposing this line into two parts.

Given a σ-finite measure space ( S, Σ, μ ), consider the Banach space L < sup > p </ sup >( μ ).

Given the fact that these gestation lengths are only estimates of an average, it is helpful to consider gestation time as a range of dates rather than a single " due date ".

Given his Anglo-American father, Lyman Marcus Warren, and American education, the Ojibwe of the time did not consider Warren one of them.

Given an-element array with an ordering relation as an input for the sorting, consider it as a collection of cards, with the ( unknown in the beginning ) statistical ordering of each element serving as its index.

Given the closeness of Diego Rivera to the people involved, and the fact that the painting is said to slightly predate the murder, some consider it to be evidence of Vidali's and Rivera's involvement in Mella's assassination, also related to Rivera's subsequent expulsion from the Mexican Communist Party.

For instance, consider the following problem: Given a directed acyclic graph, determine if the graph has a path of length 10.