x.] THE THEORY OF PROBABILITY. 209 



of heads obtained was actually 10,353, or 5 222 i* 1 the 

 first series, and 5131 in the second. The coincidence 

 with theory is pretty close, but considering the large 

 number of throws there is some reason to suspect a 

 tendency in favour of heads. 



The special interest of this trial consists in the ex- 

 hibition, in a practical form, of the results of Bernoulli's 

 theorem, and the law of error or divergence from the 

 mean to be afterwards more fully considered. It illus- 

 trates the connection between combinations and permu- 

 tations, which is exhibited in the Arithmetical Triangle, 

 and which underlies many important theorems of science. 



Probable Deductive Arguments. 



With the aid of the theory of probabilities, we may 

 extend the sphere of deductive argument. Hitherto we 

 have treated propositions as certain, and on the hypo- 

 thesis of certainty have deduced conclusions equally 

 certain. But the information on which we reason in 

 ordinary life is seldom or never certain, and almost all 

 reasoning is really a question of probability. We ought 

 therefore to be fully aware of the mode and degree in 

 which deductive reasoning is affected by the theory of 

 probability, and many persons may be surprised at the 

 results which must be admitted. Some controversial 

 writers appear to consider, as De Morgan remarked, 1 that 

 an inference from several equally probable premises is 

 itself as probable as any of them, but the true result is 

 very different. If an argument involves many proposi- 

 tions, and each of them is uncertain, the conclusion will 

 be of very little force. 



The validity of a conclusion may be regarded as a com- 

 pound event, depending upon the premises happening 

 to be true ; thus, to obtain the probability of the conclusion, 

 we must multiply together the fractions expressing the 

 probabilities of the premises. If the probability is \ that 

 A is B, and also f that B is C, the conclusion that A is C, 

 on the ground of these premises, is \ x \ or \. Similarly if 

 there be any number of premises requisite to the establish- 



1 Encyclopaedia Metropolitan^ art. Probabilities, p. 396. . 



P 



