CAUSE AND EFFECT— PROBABILITY 143 



faculties so large as compared with the percentage of 

 asserted breaches of routine, and the advantage to man- 

 kind of evolving an absolutely certain basis of knowledge 

 so great,^ that we are justified in saying that miracles 

 have been proved incredible — the word proved being used 

 in the sense in which alone it has meaning when applied 

 to the field of perceptions (p. 140). 



S I 5 . — TJie Basis of Laplace's TJieory lies in an Experience 



as to Ignorajtce 



I have said enough, I think, to indicate that if 

 Laplace's theorems be correct and can be fairly applied to 

 measure the probability of the repetition of events, our 

 belief in the routine of perceptions is based upon that high 

 degree of probability, which renders probable and prov- 

 able practically the same word. Let us consider the 

 basis of Laplace's theory a little more closely. Suppose 

 we take a shilling and toss it, then the chances that head 

 or tail will be uppermost are exactly equal ; unity de- 

 noting certainty, we say that the probability of a head 

 equals |-. If we toss it again, the chances of a head will 

 not be altered and will again be ^, and so on for each 

 throw, the chance always remaining ^. Since in two throws 

 we might with equal probability have any of the four 

 cases: head, head: tail, tail: head, tail: tail, head, it follows 

 that the recurrence of head has only a probability of ^ or 

 ^ X ^. Similarly the probability that three heads will be 

 tossed in succession may be easily seen by counting the 

 possible cases to be ^ or -^ X -g- X ^ ; that is, the odds are 

 seven to one against a triple recurrence. Extending this 

 to twenty or thirty recurrences of heads, we soon find that 

 there is an overwhelming probability against a succession 

 of recurrences without a break. 



Instead of the shilling, let us take a bag and put into 



^ This refers to the hypothesis (p. 137) that man in the course of evolu- 

 tion has attained a perceptive faculty which in the normal condition can only 

 present sequences of perceptions in the form of routine. Such routine being, 

 as we have seen, the sole basis of knowledge, is of enormous advantage to 

 man. 



