By the pigeonhole principle, any set of r + 1 vertices in the Turán graph includes two vertices in the same partition subset ; therefore,

By applying this general principle, a great number of complex compounds of osmium, ruthenium, iridium, and rhenium, with triphenylphosphine, triphenylarsine, and triphenylstibine have been obtained in this laboratory during the past few years.

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

By the same principle when the pH is too high, the kidneys excrete bicarbonate ( HCO < sub > 3 </ sub >< sup >-</ sup >) into urine as urea via the Urea Cycle ( aka the Krebs-Henseleit Ornithine Cycle ).

By the mid-19th century, however, the powerful British Empire gave up mercantilism and trade restrictions and introduced the principle of free trade, with few restrictions or tariffs.

By voluntarily stepping down and disbanding his army when the war was won, he permanently established the principle of civilian supremacy in military affairs.

By the superposition principle, the wavepacket at any time t is:

By applying the variational principle to the action

By the same principle, 10 is the least common multiple of-5 and 2 as well.

By Bernoulli's principle, this implies a pressure difference and therefore lift.

By principle, originalists are generally unwilling to defer to precedent when precedent seems to come into conflict with the Constitution.

It is for them alone to point out what we ought to do … By the principle of utility is meant that principle which approves or disapproves of every action whatsoever according to the tendency it appears to have to augment or diminish the happiness of the party whose interest is in question: or, what is the same thing in other words to promote or to oppose that happiness.

By this I mean the principle that, in deciding what is good and what is bad for a given individual, the ultimate criterion can only be his own wants and his own preferences .”

By 1865, the German physicist Rudolf Clausius had shown that this equivalence principle needed amendment.

By the principle of mathematical induction it follows that the result is true for all natural numbers.

By analyzing the bank accounts of certain members of the ring, he obtained legal proof of the principle on which the spoils had been divided.

By then the mujtahid would be able to issue his own edicts from the sources of Islamic religious laws: the Qur ' an, the Sunnah, ijmāʻ, and ' aql (" intellect ", rather than the Sunnī principle of qiyas ).

By principle, half of the inhabitants are Jewish Israelis, the rest are Muslim and Christian Arabs.

By weighting utility variations by the marginal utilities, the social welfare function implicit to the Kaldor – Hicks compensation principle is represented by anti-egalitarian, concave indifference curves.

By the equivalence principle gravitational stress-energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress-energy cannot be expressed as a non-zero tensor ; instead we have to use a pseudotensor.

By the Leonid Brezhnev period democratic centralism was described, in the 1977 Soviet Constitution, as a principle for organizing the state: " The Soviet state is organized and functions on the principle of democratic centralism, namely the electiveness of all bodies of state authority from the lowest to the highest, their accountability to the people, and the obligation of lower bodies to observe the decisions of higher ones.

By stressing prosopographical analysis, Syme rejected the force of ideas in politics, dismissing most such invocations of constitutional and political principle as nothing more than " political catchwords.

By examining " hotter " and " cooler " spark plugs of the same manufacturer side by side, the principle involved can be very clearly seen ; the cooler plugs have a more substantial ceramic insulator filling the gap between the center electrode and the shell, effectively allowing more heat to be carried off by the shell, while the hotter plugs have less ceramic material, so that the tip is more isolated from the body of the plug and retains heat better.

By the time this reaches 5 losses ( 11 rolls left ), his probability of winning on one of the remaining rolls will have dropped to ~ 50 %.

: By spectral theory, we can associate a probability measure to the values of A in any state ψ.

B is the sample space and A is an event. By the ratio of their areas, the probability of A is approximately 0. 4.

By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n tends to infinity.

By repeatedly applying the definition of conditional probability:

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By selecting the witness as a random string, the verifier is a probabilistic polynomial-time Turing Machine whose probability of accepting x when x is in X is large ( greater than 1 / 2, say ), but zero if x is not in X ( for RP ); of rejecting x when x is not in X is large but zero if x is in X ( for co-RP ); and of correctly accepting or rejecting x as a member of X is large, but zero of incorrectly accepting or rejecting x ( for ZPP ).

By repeated random selection of a possible witness, the large probability that a random string is a witness gives an expected polynomial time algorithm for accepting or rejecting an input.

By the rules of probability theory, it follows that

By contrast, for an event with a 4 in 5 probability of occurring ( i. e. a probability of 4 / 5, 0. 8 or 80 %), then the odds are 0. 8 / ( 1 − 0. 8 )

By Kolmogorov's zero-one law, for any given p, the probability that an infinite cluster exists is either zero or one.

By analogy, if in an alternate universe coins were to behave like photons and other bosons, the probability of producing two heads would indeed be one-third ( tail-head = head-tail ).

Let n ( p ; H ) be the smallest number of values we have to choose, such that the probability for finding a collision is at least p. By inverting this expression above, we find the following approximation

By the noisy-channel coding theorem, the channel capacity of a given channel is the limiting information rate ( in units of information per unit time ) that can be achieved with arbitrarily small error probability.

By use of the Black – Scholes model to value the option, one may define moneyness in terms of an ( implied ) probability distribution.

By 1643, he had been ordained a cleric in minor orders, with a benefice, and had also in all probability begun his dramatic writing.

In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f < sub > n </ sub > = f < sub > n − 1 </ sub > ± f < sub > n − 2 </ sub >, where the signs + or − are chosen at random with equal probability 1 / 2, independently for different n. By a theorem of Harry Kesten and Hillel Fürstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly.

By the Law of Large Numbers we have that, under some regularity conditions converges pointwise with probability 1 to as.

By definition, the amount of self-information contained in a probabilistic event depends only on the probability of that event: the smaller its probability, the larger the self-information associated with receiving the information that the event indeed occurred.

By Dixon's theorem, the probability that the largest factor of such a number is less than ( p − 1 )< sup > ε </ sub > is roughly ε < sup >− ε </ sub >; so there is a probability of about 3 < sup >− 3 </ sup > = 1 / 27 that a B value of n < sup > 1 / 6 </ sup > will yield a factorisation.

By staying away from complete mathematical rigor while emphasizing the physical and engineering interpretations of probability, Papoulis's book gained wide popularity.