popper-logic-scientific-discovery

154
some structural components of a theory of experience
is equal to q. (We do not assume that the distribution is equal, i.e. that
p = q.)
Now let this alternative α be at least n−1-free (n being an arbitrarily
chosen natural number). We can then ask the following question: What
is the frequency with which the property m occurs in the sequence α n?
Or in other words, what will be the value of α(n) F″(m)?
Without assuming anything beyond the fact that α is at least
n−1-free, we can settle this question 1 by elementary arithmetic. The
answer is contained in the following formula, the proof of which will
be found in appendix iii:
(1)
α F″ (m) = (n) nCmpm n − m q
The right-hand side of the ‘binomial’ formula (1) was given—in
another connection—by Newton. (It is therefore sometimes called
Newton’s formula.) I shall call it the ‘first form of the binomial
formula’.* 1
With the derivation of this formula, I now leave the frequency
theory as far as it deals with finite reference-classes. The formula will
provide us with a foundation for our discussion of the axiom of
randomness.
57 INFINITE SEQUENCES. HYPOTHETICAL
ESTIMATES OF FREQUENCY
It is quite easy to extend the results obtained for n-free finite sequences
to infinite n-free sequences which are defined by a generating period (cf.
section 55). An infinite sequence of elements playing the rôle of the
reference-class to which our relative frequencies are related may be
1 The corresponding problem in connection with infinite sequences of adjoining segments
I call ‘Bernoulli’s problem’ (following von Mises, Wahrscheinlichkeitsrechnung, 1931,
p. 128); and in connection with infinite sequences of overlapping segments I call it ‘the
quasi-Bernoulli problem’ (cf. note 1 to section 60). Thus the problem here discussed
would be the quasi-Bernoulli problem for finite sequences.
* 1 In the original text, I used the term ‘Newton’s formula’; but since this seems to be
rarely used in English, I decided to translate it by ‘binomial formula’.