If all outcomes x < sub > i </ sub > are equally likely ( that is, p < sub > 1 </ sub >

If these were the outcomes of reform, he was thankful that he had kept out of it.

If a random experiment can result in N mutually exclusive and equally likely outcomes and if N < sub > A </ sub > of these outcomes result in the occurrence of the event A, the probability of A is defined by

If in the entangled state given above Alice makes a measurement in the eigenbasis of A, there are two possible outcomes, occurring with equal probability:

If X is a real-valued random variable and a is a number then the event X ≤ a is the set of outcomes whose corresponding value of X is less than or equal to a.

If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes.

If a probability distribution depends on a parameter, one may on the one hand consider — for a given value of the parameter — the probability ( density ) of the different outcomes, and on the other hand consider — for a given outcome — the probability ( density ) this outcome has occurred for different values of the parameter.

If, in the case of education, he uses these resources to better his educational outcomes, thereby enabling him to become socially mobile, he effectively has worked to reiterate and reproduce the stratification of society, as social capital has done little to alleviate the system as a whole.

If LDL particle concentration is used to predict cardiovascular events, many other correlates of these clinical outcomes, such as diabetes mellitus, obesity and smoking, lose much of their predictive power.

* Early defibrillation-is effective for the management of ventricular fibrillation and pulseless ventricular tachycardia If defibrillation is delayed the rhythm is likely to degenerate into asystole for which outcomes are worse.

If you apply probabilities to the possible outcomes using weather forecasts or even just a calibrated probability assessment, you have quantified the uncertainty.

If this connection between our brains ' efforts to model reality and predict the future and so make possible good outcomes is disconnected from our sense of self and our will, then fatalism and self-destructive behaviors are close at hand.

If we were examining the effect of two variables, gender and premature birth, on health outcomes, we would describe any difference in health outcome scores between genders as a main effect.

If left unaddressed, hunger sets in motion an array of outcomes that perpetuate malnutrition, reduce the ability of adults to work and to give birth to healthy children, and erode children's ability to learn and lead productive, healthy, and happy lives.

If a job has a high MPS, the job characteristics model predicts that motivation, performance and job satisfaction will be positively affected and the likelihood of negative outcomes, such as absenteeism and turnover, will be reduced.

If private markets were able to provide efficient outcomes and if the distribution of income were socially acceptable, then there would be little or no scope for government.

Let the coin tosses be represented by a sequence of independent random variables, each of which is equal to H with probability p, and T with probability Let N be time of appearance of the first H ; in other words,, and If the coin never shows H, we write N is itself a random variable because it depends on the random outcomes of the coin tosses.

The system as a whole is therefore symmetric with respect to these outcomes ( If we consider any two outcomes, the probability is the same.

If you are laying bets based on Drosnin, you had better be willing to bet on all possible outcomes.

If an event is almost sure, then outcomes not in this event are theoretically possible ; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

If P is a program which outputs a string x, then P is a description of x.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.

If x is held fixed, then the Bessel functions are entire functions of α.

If the exponent r is even, then the inequality is valid for all real numbers x.

If x is a member of A, then it is also said that x belongs to A, or that x is in A.

If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If ΔS and / or T are small, the condition ΔG < 0 may imply that ΔH < 0, which would indicate an exothermic reaction.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.

If activated cytotoxic CD8 < sup >+</ sup > T cells recognize them, the T cells begin to secrete various toxins that cause the lysis or apoptosis of the infected cell.

If ADH production is excessive in heart failure, Na < sup >+</ sup > level in the plasma may fall ( hyponatremia ), and this is a sign of increased risk of death in heart failure patients.

If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.