ON DIBECT PROBABILITIES. 47 



If we add together all the probabilities of the preceding, 

 we find l -ff } or 6. This is the indication that every 

 possible event enters in six different ways into the 

 contingencies whose probabilities united amount to 

 six. 



EXAMPLE. Twelve halfpence, A x A 2 . . . A x 2 , are 

 thrown up, required the probability of all the cases 

 which can happen, and which we shall symbolise thus : 

 (HsTg) means that there are three heads and nine 

 tails. The chance of H or T in any one piece is -A; 

 consequently, the chance of the several pieces A 19 A , 

 &c., yielding each any particular letter is 1 x 4- X~-J 

 X (twelve factors), or T0 ^. Now, the 4096 pos- 

 sible cases are thus classified : 



T n ...... H n 



TT >y TT 



H 5 T 7 H 7 



H 6 T 6 happens in 924 cases. 



It appears, then, that the most likely individual re- 

 sult is He T 6 , against which, however, it is about 31 

 to 1 . But if we ask for the probability either of this 

 or of a single variation on one side or the other, that is, 

 of the following event 



Either H 5 T 7 , or H 6 T , or H 7 T 5 



wefind792 X 24-924or2508 t (m rethanhalfof4 96) 

 cases in which the event arrives. That is, the chances 

 are in favour of one or other of these arriving. As the 

 number of events increases, a given degree of nearness 

 to the most probable event becomes more and more 

 likely. To illustrate this : suppose 24 halfpence thrown 

 up, the notation remaining as before. The total number of 

 cases is now 2 24 or 1 6777216: calculating the number 



