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p times and failed q times ; what is the probability of success at the 

 p^-q-^V^^ trial ? 



This problem gives four varieties, according as m, the possible 

 number of experiments, is finite or infinite, and according as results 

 effected can or cannot be repeated. If we take the common exam- 

 ple of drawing balls, which must be either black or white, from a 

 bag, then results effected may be repeated if the balls are replaced 

 after being drawn ; if the balls di-awn are not replaced, then the 

 same result cannot be repeated. The only case of the problem which 

 the author of this paper has been able to find solved in any treatise 

 on probabilities, though he must confess that his range of inquiry 

 has not been very large, is that where m is infinite, and the balls 

 drawn are replaced. His object in his paper was to solve the case 

 where m is finite, and the balls are not replaced. 



In this case it is manifest that the number of white balls con- 

 tained in the bag may be any number from m — q to p, and the cor- 

 responding number of black, any number from q to m—p. Then, 

 omitting the common constants, the several hypotheses which can be 

 formed as to the proportion of black and white balls in the bag at 

 first, Hj, Hg, Hg, &c., give, for the probability of the event ob- 

 served, that is, for the drawing of p white and q black balls, the fol- 

 lowing probabilities : — 



Hj, m — q . m — q — \ m — q—p+ 1x1.2.3 q (a) 



Hg, m—q— I . m — q — 2 m — q—p x 2 . 3 . 4 9+1 (jS) 



and so on for all the other hypotheses. Hence the probability of H^, 



which is ;^ — &c. = (by the preceding summation) 



a + i8 + 7 \ J If o J 



p + 1 .p + 2 p + g, + l 



m + 1 . m m— j3 — 9+ 1x1.2.3 q 



m — q . m—q—\ m — q — p+ 1 x 1 . 2 q. 



Therefore the probability of a white ball Ai p + q + V^ drawing, de- 

 rived fi'om H^, is 



p + 1 .p + 2 pJrq+l 



(m+ 1 . m m—p — q-^ l)xl.2.3 q 



m — q . m — q — 1 m — q — p x 1 . 2 . 3 



X 



