SECT. 11.] Modes of establishing the Groups or Series. 85 



number of throws we should probably begin to find, under 

 such circumstances, that either head or tail was having a 

 preference shown to it. If so, would not similar effects be 

 found to be connected with the way in which we started 

 each successive pair of throws ? According as we chose to 

 make a practice of putting HH or TT uppermost, might 

 there not be a disturbance in the proportion of successions of 

 two heads or two tails ? Following out this train of reason 

 ing, it would seem to point with some likelihood to the con 

 clusion that in order to obtain a series of the kind we 

 expect, we should have to dispose the antecedents in a 

 similar series at the start. The changes and chances pro 

 duced by the act of throwing might introduce infinite in 

 dividual variations, and yet there might be found, in the very 

 long run, to be a close similarity between these two series. 



11. This is, to a certain extent, only shifting the dif 

 ficulty, I admit; for the claim formerly advanced about the 

 possibility of proving the proportions of the throws in the 

 former series, will probably now be repeated in favour of 

 those in the latter. Still the question is very much nar 

 rowed, for we have reduced it to a series of voluntary acts. 

 A man may put whatever side he pleases uppermost. He 

 may act consciously, as I have said, or he may think nothing 

 whatever about the matter, that is, throw at random; if so, 

 it will probably be asserted by many that he will involun 

 tarily produce a series of the kind in question. It may be 

 so, or it may not; it does not seem that there are any 

 easily accessible data by which to decide. All that I am 

 concerned with here is to show the likelihood that the com 

 monly received result does in reality depend upon the ful 

 filment of a certain condition at the outset, a condition which 

 it is certainly optional with any one to fulfil or not as he 

 pleases. The short successions doubtless will take care of 



