popper-logic-scientific-discovery

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some structural components of a theory of experience
small fixed fraction (which we may freely choose). We can then say
that the probability of chancing upon a fair sample approaches 1 as
closely as we like if only we make the segments in question sufficiently
long.* 1
In this formulation the word ‘probability’ (or ‘value of the probability’)
occurs twice. How is it to be interpreted or translated here? In the sense
of my frequency definition it would have to be translated as follows (I
italicize the two translations of the word ‘probability’ into the frequency
language): The overwhelming majority of all sufficiently long finite
segments will be ‘fair samples’; that is to say, their relative frequency
will deviate from the frequency value p of the random sequence in question
by an arbitrarily fixed small amount; or, more briefly: The frequency
p is realized, approximately, in almost all sufficiently long segments.
(How we arrive at the value p is irrelevant to our present discussion; it
may be, say, the result of a hypothetical estimate.)
Bearing in mind that the Bernoulli frequency αn F(∆p) increases
monotonically with the increasing length n of the segment and that it
decreases monotonically with decreasing n, and that, therefore, the
value of the relative frequency is comparatively rarely realized in short
segments, we can also say:
Bernoulli’s theorem states that short segments of ‘absolutely free’ or
chance-like sequences will often show relatively great deviations from p
and thus relatively great fluctuations, while the longer segments, in
most cases, will show smaller and smaller deviations from p with
increasing length. Consequently, most deviations in sufficiently long
segments will become as small as we like; or in other words, great
deviations will become as rare as we like.
Accordingly, if we take a very long segment of a random sequence,
in order to find the frequencies within its sub-sequences by counting,
or perhaps by the use of other empirical and statistical methods, then
we shall get, in the vast majority of cases, the following result. There is
a characteristic average frequency, such that the relative frequencies in
the whole segment, and in almost all long sub-segments, will deviate
* 1 This sentence has been reformulated (without altering its content) in the translation
by introducing the concept of a ‘fair sample’: the original operates only with the
definiens of this concept.