1 to 1 ½ ½
2 to 1 ⅔ ⅓
3 to 1 ¾ ¼
3 to 2 ⅗ ⅖
4 to 1 ⅘ ⅕
4 to 3 ⁴⁄₇ ³⁄₇
5 to 1 ⅚ ⅙
&c. &c. &c.
Let the probability of a conclusion, as derived from the premises (that
is on the supposition that it was never imagined to be possible till the
argument was heard), be called the _intrinsic probability_ of the
argument. This is found by multiplying together the probabilities of all
the assertions which are necessary to the argument. Thus, suppose that a
conclusion was held to be impossible until an argument of a single
syllogism was produced, the premises of which have severally five to one
and eight to one in their favour. Then ⅚ × ⁸⁄₉, or ⁴⁰⁄₅₄, is the
intrinsic probability of the argument, and the odds in its favour are 40
to 14, or 20 to 7.
But this intrinsic probability is not always that of the conclusion; the
latter, of course, depending in some degree on the likelihood which the
conclusion was supposed to have before the argument was produced. A
syllogism of 20 to 7 in its favour, advanced in favour of a conclusion
which was beforehand as likely as not, produces a much more probable
result than if the conclusion had been thought absolutely false until
the argument produced a certain belief in the possibility of its being
true. The change made in the probability of a conclusion by the
introduction of an argument (or of a new argument, if some have already
preceded) is found by the following rule.
From the sum of the existing probability of the conclusion and the
intrinsic probability of the new argument, take their product; the
remainder is the probability of the conclusion, as reinforced by the
argument. Thus, _a + b − ab_ is the probability of the truth of a
conclusion after the introduction of an argument of the intrinsic
probability _b_, the previous probability of the said conclusion having
been _a_.
Thus, a conclusion which has at present the chance ⅔ in its favour, when
reinforced by an argument whose intrinsic probability is ¾, acquires the
probability ⅔ + ¾ − ⅔ × ¾ or, ⅔ + ¾ − ½, or ¹¹⁄₁₂; or, having 2 to 1 in
its favour before, it has 11 to 1 in its favour after, the argument.
When the conclusion was neither likely nor unlikely beforehand (or had
the probability ½), the shortest way of applying the preceding rule (in
which _a + b − ab_ becomes ½ + ½_b_) is to divide the sum of the
numerator and denominator of the intrinsic probability of the argument
by twice the denominator. Thus, an argument of which the intrinsic
probability is ¾, gives to a conclusion on which no bias previously
existed, the probability ⅞ or (3 + 4)/(2 × 4).