kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

compile incompatible, solely and equally possible propositions 1 , 2 , …, , µ of whichhave B as their join. But it is obvious that d i ~ , k because, when assuming for example that d 1> 1 , we would have obtained that always d i < i and, in accord with Axiom 2.2b, > which is impossible. But then, d 1 ~ 1 , d 2 ~ 2 etc, hence(d 1 or d 2 or … or d µ ) ~ ( 1 or 2 or … or µ );that is, A ~ B, QED.2.1.4. Definition of Mathematical ProbabilityThe coefficient, which we called the mathematical probability of A, is thus quitedetermined by the fraction m/n where n is the number of solely and equally possibleincompatible propositions m of which have A as their join. Consequently, this coefficient is afunction of m/n which we denote by (m/n). On the basis of the above, should beincreasing and this necessary condition is at the same time sufficient for satisfying all theassumed axioms if only the function (m/n) can be fixed once and forever for all thetotalities that might be added to the given one. Since such a function can be chosenarbitrarily, we assume its simplest form: (m n) = m/n so that m/n is called the mathematicalprobability of A.However, in accord with the main axioms we could have just as well chosen m 2 /n 2 , m/(n –m), etc. The assumption of one or another verbal definition of probability would haveobviously influenced the conclusions of probability theory just as little as a change of a unitof measure influences the inferences of geometry or mechanics. Only the form but not thesubstance of the theorems would have changed; we would have explicated the theory ofprobability in a new terminology rather than obtained a new theory. The agreement that I amintroducing here is therefore of a purely technical nature 7 as contrasted with the case of themain axioms assumed above and characterizing the essence of the notion of probability: theirviolation would have, on the contrary, utterly changed the substance of probability theory.Note. Together with Borel (1914, p. 58) we might have called the fraction m/(n – m), – theratio of the number of the favorable cases to the number of unfavorable cases; or, theexpression (m/n )/[(n – m)/n], – the ratio of the probability of a proposition to that of itsnegation, – the relative probability of the proposition.Remark. When adding a new totality to the given one, we must, and we can alwaysdistribute, in accord with the axioms, the values of the probabilities of the newly introducedpropositions in such a manner that the given propositions will still have the same probability{probabilities} as they had in the original totality. Indeed, suppose that the elementarypropositions A 1 , A 2 , …, A n in the given totality are equally possible; consequently, afterchoosing the function (m/n) all the propositions of the totality acquire quite definite values.Add the second totality formed of elementary propositions B 1 , B 2 , …, B k and let us agree toconsider that, for example, all the combinations (A i and B j ) in the united totality are alsoequally possible. If the function persists all the propositions of the united totality willobtain definite probabilities and any join of the type (A 1 or A 2 or … or A m ) previously havingprobability (m/n) and regarded as a join[(A 1 and B 1 ) or (A 1 and B 2 ) or … or (A m and B k )]must now have probability (km/kn) equal to its previous value, (m/n). And all thepropositions B j will also be equally possible.We may thus agree to consider any incompatible and solely possible propositionsA 1 , A 2 , …, A k (16)