popper-logic-scientific-discovery

segments possessing the property ‘∆p’—within the α n-sequences; it
thus answers the question of the value of αn F(∆p).
Intuitively one might guess that if the value δ (with δ > 0) is fixed,
and if n increases, then the frequency of these segments with the property
∆p, and therefore the value of αn F(∆p), will also increase (and
that its increase will be monotonic). Bernoulli’s proof (which can be
found in any textbook on the calculus of probability) proceeds by
evaluating this increase with the help of the binomial formula. He finds
that if n increases without limit, the value of αn F(∆p) approaches
the maximal value 1, for any fixed value of δ, however small. This may
be expressed in symbols by
lim
n →∞ αnF(∆p) = 1 (for any value of ∆p) (1)
This formula results from transforming the third binomial formula
for sequences of adjoining segments. The analogous second binomial formula
for sequences of overlapping segments would immediately lead, by
the same method, to the corresponding formula
lim
n →∞ α(n)F′(∆p) = 1 (2)
which is valid for sequences of overlapping segments and normal
ordinal selection from them, and hence for sequences with after-effects
(which have been studied by Smoluchowski 2 ). Formula (2) itself
yields (1) in case sequences are selected which do not overlap, and
which are therefore n-free. (2) may be described as a variant of
Bernoulli’s theorem; and what I am going to say here about Bernoulli’s
theorem applies mutatis mutandis to this variant.
Bernoulli’s theorem, i.e. formula (1), may be expressed in words as
follows. Let us call a long finite segment of some fixed length, selected
from a random sequence α, a ‘fair sample’ if, and only if, the frequency
of the ones within this segment deviates from p, i.e. the value of the probability
of the ones within the random sequence α, by no more than some
2 Cf. note 3 to section 60, and note 5 to section 64.
probability 169