AND ITS APPLICATION TO A PROBLEM IN PROBABILITIES. 543 
All that we can say is, that experience has led dispassionate men to come to nearly 
the same Pa as the mathematician: for while he asserts the probability 

of success to be |, a they act upon the supposition that the probabilities of 
success and ephire are proportioned to the number of experienced cases of success 
and failure : and when either p or g isa large number, that is, when the experience 
is great, the conclusion and the supposition coincide. 
In order to realise the Problem, we shall use the ordinary illustration, and 
suppose that a bag contains m balls in unknown proportions of black and white, 
but all either black or white; that p white and g black balls have been drawn, 
and that it is required to find the probability of drawing a white at the p+q+1'™ 
drawing. 
The Problem as thus stated, admits of four varieties. 
1. m may be given, and the balls drawn may have been replaced in the bag. 
2. m may be given, and the balls drawn not replaced. 
: 3. m may be infinite or indefinite, and the balls replaced. 
- 4. m may be infinite or indefinite, and the balls not replaced. 
Of these, the 3d is the only case which I find solved in the treatises which I 
have consulted. I propose to solve the 2d case, and therein the 4¢h; and, in 
conclusion, to make an attempt at the solution of the 1st case. 
To render the observed event, that is, the drawing of p white and q black 
balls (or E), possible, the original number of whites may have been any number 
from m—g to p inclusive, and the number of blacks any number from g to m—p. 
Let us call the hypothesis of m—g white and q black, H, 


and m—gq-—1 white and g+1 black, H,, &c. 
m— oF m—gqa=1 .....m—q—pt1x1.2.3.....g* 
Then H, gives for probability of E MSL ne BA 
or, calling the denominator A, 
H, gives 5 -m—q.m—q-1 atch m—q—p+1x1.2.3..... q (a) 
; peta tee Ee m—q—px2.3.4....q+1 (8) } (F) 
iH: gives m—q—2.m—q—3.... m—gq—p—1x3.4....q4+2 (yy) 
- &e. &e. 
_ Now, a+8++, &c. by the former proposition (E) 
EIS SG cS ae ee 1 BSOne 
> Re oeT pot RPC aa el ria p—qt+l 
-. probability of oS i Baye Soy. Ges 

* The coefficient (U of Gattoway’s Treatise), expressing the number of different ways in which 
p white and gq black balls can be combined in p+q trials, is here omitted. This is immaterial, 
as it disappears in the expression magi ks 
