SECT. 16.] On certain kinds of Groups or Series. 21 j 



2five anything approaching to 50 for such a chance. Pro- 

 bably not, because no man would see enough of the series to 

 make it worth his while. What most persons form their 

 practical opinion upon, is such small portions of the series 

 as they have actually seen or can reasonably expect. Now 

 in any such portion, say one which embraces 100 turns, the 

 longest succession of heads would not amount on the average 

 to more than seven or eight. This is observed, but it is for 

 gotten that the formula which produced these, would, if it 

 had greater scope, keep on producing better and better ones 

 without any limit. Hence it arises that some persons are 

 perplexed, because the conduct they would adopt, in reference 

 to the curtailed portion of the series which they are practically 

 likely to meet with, does not find its justification in inferences 

 which are necessarily based upon the series in the complete 

 ness of its infinitude. 



16. This will be more clearly seen by considering the 

 various possibilities, arid the scope required in order to exhaust 

 them, when we confine ourselves to a limited number of 

 throws. Begin with three. This yields eight equally likely 

 possibilities. In four of these cases the thrower starts with 

 tail and therefore loses : in two he gains a single point 

 (i.e. 1); in one he gains two points, and in one he gains four 

 points. Hence his total gain being eight pounds achieved in 

 four different contingencies, his average gain would be two 

 pounds. 



Now suppose he be allowed to go as far as n throws, so 

 that we have to contemplate 2 n possibilities. All of these 

 have to be taken into account if we wish to consider what 

 happens on the average. It will readily be seen that, when 

 all the possible cases have been reckoned once, his total gain 

 will be (reckoned in pounds), 



2-2 + 2 n ~ 3 . 2 + 2 n ~ 4 . 2 2 + . . . . -f 2 . 2 n ~ 3 + 2 n ~ 2 + 2 n-1 , 



