popper-logic-scientific-discovery

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new appendices
(3) It will be applicable, with slight modifications, whenever we
can assign a weight function to the various possibilities.
(4) It will be applicable, or it will be of heuristic value, in most
cases where an over-simplified estimate that works with equal possibilities
leads to a solution approaching to the probabilities zero or one.
(5) It will be of great heuristic value in cases in which weights can
be introduced in the form of probabilities. Take, for example, the following
simple problem: we are to calculate the probability of throwing
with a die an even number when the throws of the number six are not
counted, but considered as ‘no throw’. The classical definition leads, of course,
to 2/5. We may now assume that the die is biased, and that the
(unequal) probabilities p(1), p(2), . . . , p(6) of its sides are given. We
can then still calculate the required probability as equal to
p(2) + p(4)
p(1) + p(2) + p(3) + p(4) + p(5)
= p(2) + p(4)
1 − p(6)
That is to say, we can modify the classical definition so as to yield the
following simple rule:
Given the probabilities of all the (mutually exclusive) possible cases,
the required probability is the sum of the probabilities of all the
(mutually exclusive) favourable cases, divided by the sum of the
probabilities of all the (mutually exclusive) possible cases.
It is clear that we can also express this rule, for exclusive or
non-exclusive cases, as follows.
The required probability is always equal to the probability of the
disjunction of all the (exclusive or non-exclusive) favourable cases,
divided by the probability of the disjunction of all the (exclusive or
non-exclusive) possible cases.
(6) These rules can be used for a heuristic derivation of the definition
of relative probability, and of the general multiplication theorem.
For let us symbolize, in the last example, ‘even’ by ‘a’ and ‘other than
a six’ by ‘b’. Then our problem of determining the probability of an
even throw if we disregard throws of a six is clearly the same as the
problem of determining p(a, b), that is to say, the probability of a, given
b, or the probability of finding an a among the b’s.
The calculation can then proceed as follows. Instead of writing