_ 



&amp;gt;2 On certain kinds of Groups or Series. [CHAP. I. [ 



Viz. (n + 1) 2 n 2 . 



This being spread over 2 n-1 different occasions of gain his 



average gain will be %(n + 1). 



Now when we are referring to averages it must be re 

 membered that the minimum number of different occurrences 

 necessary in order to justify the average is that which enables I 

 each of them to present itself once. A man proposes to stop 

 short at a succession of ten heads. Well and good. We tell 

 him that his average gain will be 5. 10s. Od.: but we also 

 impress upon him that in order to justify this statement he 

 must commence to toss at least 1024 times, for in no less 

 number can all the contingencies of gain and loss be exhibited 

 and balanced. If he proposes to reach an average gain of 

 20, he will require to be prepared to go up to 39 throws. 

 To justify this payment he must commence to throw 2 39 

 times, i.e. about a million million times. Not before he has 

 accomplished this will he be in a position to prove to any 

 sceptic that this is the true average value of a turn extend 

 ing to 39 successive tosses. 



Of course if he elects to toss to all eternity we must I 

 adopt the line of explanation which alone is possible where :; 

 questions of infinity in respect of number and magnitude are if 

 involved. We cannot tell him to pay down an infinite sum/ H 

 for this has no strict meaning. But we tell him that, however | 

 much he may consent to pay each time runs of heads occur, \ 

 he will attain at last a stage in which he will have won 

 back his total payments by his total receipts. However 

 large n may be, if he perseveres in trying 2 n times he may 

 have a true average receipt of (n + 1) pounds, and if he 

 continues long enough onwards he will have it. 



The problem will recur for consideration in a future 

 chapter. 



