kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

For the time being, we leave aside the issue of whether there exists such objectiveinformation that any person will agree that they predetermine the events A and B to the sameextent, so that they should be equally expected, should be considered equally possible.However, even when denying the availability of such information for each case, each personwho attempts to understand to what extent he may count on the occurrence of some event,the following axioms of §2.1 will be compulsory.1. We ought to reckon on a certain event more than on an uncertain event.2. If we expect A and A 1 to the same extent; if, further, the same is true with respect to Band B 1 ; if A is incompatible with B, and A 1 incompatible with B 1 , – then we should equallyexpect (A or B) and (A 1 or B 1 ). On the contrary, if we expect B rather than B 1 , then we expect(A or B) more than (A 1 or B 1 ).The Axiom of realization 2.3 (§2.2.3) will become just as obvious, if, when stating it, weattach the abovementioned sense to the notion of probability: If and are particular casesof A and B respectively, we should, when counting on A as much as on B, equally expect theoccurrence of and ; and, if A occurs, we should expect to the same extent as providedthat B occurs.Depending on whether our assumptions of the equal probability of the considered eventsare objective or subjective, the conclusions, following from our objectively compulsory (for anormal state of mind) axioms and theorems, will acquire an objective or a more or lesssubjective meaning.We ought to show now that the assumptions about an equal possibility of two phenomenacan be as objective as the premise of the equality of any two concrete quantities whatsoever;and to reveal thus the scientific importance of the theory of probability.4.3. EquipossibilityTo this end, let us consider an example. A homogeneous sphere is placed on a cylinder ofrevolution with horizontal elements in such a manner that its center {of gravity}is on thesame vertical line with the point of contact. Had the experiment been realized ideally, thesphere would have been in equilibrium. However, it follows both from mechanics andexperience that this equilibrium is unstable. An unyielding to measurement deviation fromthe conditions of an ideal experiment is sufficient for the sphere to roll to one or to the otherside. If the practitioner will realize this experiment with all the possible precision by takingall measures for the deviation of the cylinder to one side not to outweigh its deviation to theother side, the outcome would have remained unknown to him. It is naturally possible thatanother experimentalist with more precise instruments can foresee the result; but then heshould again modify the experiment for arriving at the same pattern of unstable equilibrium,and the new outcome would be just as unknown to him as it was to his predecessor.When preparing the second experiment identical with the first one as much as possible, ourpractitioner will have the same grounds for expecting a similar result. Had the experimentbeen stable with its outcome not being influenced by such differences in its arrangement,which are not allowed for, we might have foreseen that the results in both cases would be thesame. However, owing to the mechanical instability of the realized layout, we restrict ourstatement by concluding that a definite outcome of the second experiment (the movement ofthe sphere to the right) has the same probability as in the first one.In general, if the difference between the causes leading to the occurrence of events A andB is so negligible as not to be detected or measured, these events are recognized as equallyprobable.Once this definition is admitted, it also directly leads to the previously assumed axioms.However, our axiomatic construction of the theory of probability is not connected withaccepting or disregarding it. An absolute equality of probabilities naturally represents only amathematical abstraction, just as the equality{congruence}of segments does; for establishing