40 ESSAY ON PROBABILITIES. 



or the general chances which any set of circumstances 

 affords, if I multiply the chances for an event any num- 

 ber of times, provided I multiply the chances against it 

 as often. Reduce the preceding lotteries to common num- 

 bers of balls, by putting 20 balls for one into the second, 

 and 9 for one into the first. We then have 

 (90 white, 90 red) (80 black, 100 red). 

 Now mix the two, and we have 



(90 white, 80 black, 190 red) : 



and the chance of a white is -$-$ 9 or -i, that is, the pro- 

 blem is not altered by the mixture. And the same may 

 be shown in any other case. 



Let us now suppose that the chance of A is and 

 that of B . The chances against A and B are there- 

 fore J- and -|. 



The possible cases are The probability of which is 

 That A and B shall both happen x f or ^ 



That A shall happen, and not B | x | or 5 \ 



That B shall happen, and not A $ x f or 5 5 T 



That neither shall happen. x f or 5 2 T 



One of these cases must happen, and the sum of the 

 chances is -T or 1, each compound event being ex. 

 elusive of the others. Again, we find 



(~Both or neither is 

 One or other, but only one , 

 One or other, or both 



B or both } B or neither 5 \ 

 j A or neither, or both i| 

 ^B or neither, or both || 



The latter set of events does not consist of those which 

 are mutually exclusive. Any thing which happens falls 

 under six of them, that is, six of them must happen. 

 Thus, A, and not B, actually arriving, would secure any 

 gain which depended upon either of the following : 



One or other, but only one A, or both 



One or other, or both A, or neither 



One or neither A, or neither, or both, 



