popper-logic-scientific-discovery

(5)
Now let h be the statement
P(a, b) = r
and let e be the statement ‘In a sample which has the size n and which
satisfies the condition b (or which is taken at random from the population
b), a is satisfied in n(r ± δ) of the instances’.* 1 Then we may put,
especially for small values of δ,
(6)
P(e) ≈ 2 δ* 2
appendix *ix 429
We may even put P(e) = 2δ; for this would mean that we assign equal
probabilities—and therefore, the probabilities 1/(n + 1)—to each of
the n + 1 proportions, 0/n, 1/n, ... n/n, with which a property a may
occur in a sample of the size n. It follows that we should have to assign
the probability, P(e) = (2d + 1)/(n + 1), to a statistical report e telling
us that m ± d members of a population of the size n have the property a;
so that putting δ = (d + (1/2))/(n + 1), we obtain P(e) = 2δ. (The
equidistribution here described is the one which Laplace assumes in
the derivation of his rule of succession. It is adequate for assessing the
absolute probability, P(e), if e is a statistical report about a sample. But it is
inadequate for assessing the relative probability P(e, h) of the same
report, given a hypothesis h according to which the sample is the
product of an n times repeated experiment whose possible results occur
each with a certain probability. For in this case, it is adequate to assume
a combinatoric, i.e. a Bernoullian rather than a Laplacean, distribution.)
We see from (6) that, if we wish to make P(e) small, we have to make
δ small.
On the other hand, P(e, h)—the likelihood of h—will be close to 1
* 1 It is here assumed that if the size of the sample is n, the frequency within this sample
can be determined at best with an imprecision of ± 1/2n; so that for a finite n, we have
δ � 1/2n. (For large samples, this simply leads to δ > 0.)
* 2 Formula (6) is a direct consequence of the fact that the informative content of a
statement increases with its precision, so that its absolute logical probability increases
with its degree of imprecision; see sections 34 to 37. (To this we have to add the fact that
in the case of a statistical sample, the degree of imprecision and the probability have the
same minima and maxima, 0 and 1.)