33 ESSAY ON PROBABILITIES. 



VII. is to be drawn, according as III. gives W or B. 

 What are the probabilities of the several possible 

 drawings ? 



If from I. we draw W (of which the chance is ), 

 we proceed to II. If we still draw W (chance, -J), we 

 proceed to IV. And here the chance of W is -J-. 

 Hence, 



the chance of WWW is f x x \ = 2 3 5 



Computing all the other chances in the same way, we 

 get the following : 



1. WWW | f i = A 



2. WWB | i - ft 



3. WBW f = ft 

 4.WBB f | J-A 



5. BWW f J J = A 



6. BWB | M | 



7. BBW f j = ft 



8. BBB | \ I = fa 



The sum of all these is equal to unity, as it should 

 be, since one or other of these cases must happen. And 

 by reducing all to the common denominator 5. 5. 4. 6, 

 or 600, we have the following chances : 



'7. BBWfif 



1. WWW, 



2. WWB , 



3. \VBW 



5. BWW 



6. BWB 



That all shall he white, 7 to 1 against (nearly). 



Two ^vhite and one black, 3 to 1 against (very nearly) 



Two black and one white, an even chance (nearly) 



All black, 10 to 1 against (nearly). 



From what has been said in this chapter, no great 

 difficulty will be found in ordinary questions. The cir- 

 cumstances are supposed to be fully known, and the 

 probabilities will be found, of the strength which it 

 follows they must have, to those who admit the axioms 

 on which the measure of probability is founded. 



