popper-logic-scientific-discovery

388
new appendices
All this does not mean that we cannot express the difference in
content between a 1 and a 2 in terms of probability, at least in some
cases. For example, the fact that a 1 entails a 2 but not vice versa would give
rise to
p(a 1, a 2) = 0; p(a 2, a 1) = 1
even though we should have, at the same time, p(a 1) = p(a 2) = 0.
Thus we should have
p(a 1, a 2) ’ and ‘ < ’. (We may also use ‘ ’, or ‘is higher or equally high’,
p(a 1a 2 ... a n) must tend to zero provided all the a i are mutually independent. Thus the
probability of tossing n successive heads is, according to all probability theories, 1/2 n ,
which becomes zero if the number of throws becomes infinite.
A similar problem of probability theory is this. Put into an urn n balls marked with the
numbers 1 to n, and mix them. What is the probability of drawing a ball marked with a
prime number? The well-known solution of this problem, like that of the previous one,
tends to zero when n tends to infinity; which means that the probability of drawing a ball
marked with a divisible number becomes 1, for n →∞, even though there is an infinite
number of balls with non-divisible numbers in the urn. This result must be the same in
any adequate theory of probability. One must not, therefore, single out a particular theory
of probability, such as the frequency theory, and criticize it as ‘at least mildly paradoxical’
because it yields this perfectly correct result. (A criticism of this kind will be
found in W. Kneale’s Probability and Induction, 1949, p. 156). In view of our last ‘problem of
probability theory’—that of drawing numbered balls—Jeffrey’s attack on those who
speak of the ‘probability distribution of prime numbers’ seems to me equally unwarranted.
(Cf. Theory of Probability, 2nd edition, p. 38, footnote.)