122 REMARKS ON THE DOCTRINES OF CHANCli, 



being apprehensive from your having declined favouring us 

 with your reraiarks, that your view of the subject might not 

 be very dissimilar to your correspondent's. 

 The case ob- Opsimath admits de Moivre's tirst case, viz. that if any 

 jected to. ^^^ were to undertake to throw an ace in one throw with one 



die, he would have ^ of all the possible chances in his fa- 

 vour, aad the remaining f against him : but objects to the 

 second case, viz. that, if he were to undertake to do it in 

 two throws with one die (or, which is certainly the same 

 thing, in one throw with 2 dice) that the chances in his fa- 

 vour are -^^ and f |- against it ; alleging as a reason, that two 

 equal chances are twice as good as one, atid that of course 

 Attempt to de- it should be if instead of 3^. This reasoning is correct if 

 it. -j-j^g chances are of equal value : but this I apprehend is not 



the case, the second chance being less than the first by the 

 probability of the first's succeeding; and as a confirmation 

 of de Moivre's doctrine being correct, it appears by the fol- 

 lowing statement, that of all the 36 possible combinations 

 with 2 dice, there are but 11 throws which give an ace or 

 any other particular number. Let 1 die be called A, the 

 other B ; then may be thrown with 



A 



B 



A 



B 



A 



B 



A 



B 



A 



B 



A 



B 





1 



2 



1 



3 



1 





1 



5 



1 



6 



1 





2 



2 



2 



3 



2 





2 



5 



2 



6 



2 





3 



2 



3 



3 



3 





3 



5 



3 



6 



3 





4 



2 



4 



3 



4 





4 



5 



4 



6 



4 





5 



2 



5 



3 



5 





5 



5 



5 



6 



5 





6 



2 



6 



3 



6 



4 



6 



5 



6 



6 



6 



and again, as the chance of throwing an ace with one die is 

 admitted by your correspondent to be ^, and of not doing 

 it ^y the chance of not doing it with either of two dice is 

 1^ X i 3Z ff , and this subtracted from unity, which repre- 

 sents the certainty of an ace being either thrown or not 

 thrown, gives 4-Jj as above. 

 The argument Opsimath farther objects to this statement, and says, if 

 puisued. ^g proceed according to the above method, the probability 



of throwing an ace with one die in 6 throws does not amount 

 to f, or a certainty. Nor should it: for were this the case, 

 he might undertake to pay any sunij provided he did not 



do 



