20 On certain kinds of Groups or Series. [CHAP. I. 



than that number of times. Here then is a series formed by 

 a succession of throws. We will assume, what many per 

 sons will consider to admit of demonstration, and what : 

 certainly experience confirms within considerable limits, 

 that the rarity of these runs of the same face is in direct 

 proportion to the amount I receive for them when they d&amp;lt; 

 occur. In other words, if we regard only the occasions on 

 which I receive payments, we shall find that every othei 

 time I get one pound, once in four times I get two pounds 

 once in eight times four pounds, and so on without any end 

 The question is then asked, what ought I to pay for this 

 privilege ? At the risk of a slight anticipation of the results 

 of a subsequent chapter, we may assume that this is equiva 

 lent to asking, what amount paid each time would on th( 

 average leave me neither winner nor loser ? In other words 

 what is the average amount I should receive on the above 

 terms ? Theory pronounces that I ought to give an infinite 

 sum: that is, no finite sum, however great, would be an 

 adequate equivalent. And this is really quite intelligible 

 There is a series of indefinite length before me, and th( 

 longer I continue to work it the richer are my returns, am 

 this without any limit whatever. It is true that the very 

 rich hauls are extremely rare, but still they do come, and 

 when they come they make it up by their greater richness 

 On every occasion on which people have devoted themselve, 

 to the pursuit in question, they made acquaintance, of course 

 with but a limited portion of this series ; but the series on 

 which we base our calculation is unlimited; and the in 

 ferences usually drawn as to the sum which ought in the long 

 run to be paid for the privilege in question are in perfed 

 accordance with this supposition. 



The common form of objection is given in the reply, thai 

 so far from paying an infinite sum, no sensible man woulc 



