popper-logic-scientific-discovery

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some structural components of a theory of experience
that does not vary with the ordinal number of the element). And it
follows, finally, that ‘absolute freedom’ will entail insensitivity to all* 3
combinations of these two types of selection.
61 THE LAW OF GREAT NUMBERS
(BERNOULLI’S THEOREM)
Bernoulli’s theorem, or the (first1 ) ‘law of great numbers’ can be
derived from the third binomial formula by purely arithmetical reasoning,
under the assumption that we can take n to the limit, n →∞. It can
therefore be asserted only of infinite sequences α; for it is only in these
that the n-segments of αn-sequences can increase in length indefinitely.
And it can be asserted only of such sequences α as are ‘absolutely free’,
for it is only under the assumption of n-freedom for every n that we can
take n to the limit, n →∞.
Bernoulli’s theorem provides the solution of a problem which is
closely akin to the problem which (following von Mises) I have called
‘Bernoulli’s problem’, viz. the problem of the value of F(m). As
αn
indicated in section 56, an n-segment may be said to have the property
‘m’ when it contains precisely m ones; the relative frequency of ones
within this (finite) segment is then, of course, m/n. We may now
define: An n-segment of α has the property ‘∆p’ if and only if the
relative frequency of its ones deviates by less than δ from the value
αF(1) = p, i.e. the probability of ones in the sequence α; here, δ is any
small fraction, chosen as near to zero as we like (but different from
zero). We can express this condition by saying: an n segment has the
property ‘∆p’ if and only if | m
− p |