popper-logic-scientific-discovery

164
some structural components of a theory of experience
60 BERNOULLI’S PROBLEM
The first binomial formula which was mentioned in section 56, viz.
α (n) F″ (m) = n C mp m q n − m (1)
holds for finite sequences of overlapping segments. It is derivable on
the assumption that the finite sequence α is at least n−1-free. Upon the
same assumption, we immediately obtain an exactly corresponding
formula for infinite sequences; that is to say, if α is infinite and at least
n−1-free, then
α (n) F′ (m) = n C mp m q n − m (2)
Since chance-like sequences are absolutely free, i.e. n-free for every n,
formula (2), the second binomial formula, must also apply to them; and
it must apply to them, indeed, for whatever value of n we may choose.
In what follows, we shall be concerned only with chance-like
sequences, or random sequences (as defined in the foregoing section).
We are going to show that, for chance-like sequences, a third binomial
formula (3) must hold in addition to formula (2); it is the formula
α n F(m) = n C mp m q n − m (3)
Formula (3) differs from formula (2) in two ways: First, it is asserted
for sequences of adjoining segments α n instead of for sequences of
overlapping segments α (n). Secondly, it does not contain the symbol F′
but the symbol F. This means that it asserts, by implication, that the
sequences of adjoining segments are in their turn chance-like, or random; for F,
i.e. objective probability, is defined only for chance-like sequences.
The question, answered by (3), of the objective probability of the
property m in a sequence of adjoining segments—i.e. the question of
the value of αn F(m)—I call, following von Mises, ‘Bernoulli’s problem’.
1 For its solution, and hence for the derivation of the third
1 The corresponding question for sequences of overlapping segments, i.e. the problem of
α (n) F′(m), answered by (2), can be called the ‘quasi-Bernoulli problem’; cf. note 1 to
section 56 as well as section 61.