XX. On a Question in the Theory of Probabilities. By Augustus 

 De Morgan, of Trinity College, Professor of Mathematics in 

 University College, Lotidon. 



[Read February 26, 183/.] 



The object of this paper is the correction of an oversight made 

 both by Laplace and M. Poisson, in pages 279 and 209 of their 

 respective works on The Theory of Probabihties. 



The reputation of neither of those analysts requires an explanatory 

 eulogium to accompany the detection of an error in their writings, 

 particularly on a subject so liable to cause mistake as the theory in 

 question : I shall therefore proceed at once to the point. Both arrive 

 correctly at the conclusion, that* 



2 rtj^ 



^ f^'e-'\lt ^ -^ (1) 



represents the probability that the number of arrivals of A shall fall 

 between v — I and v + 1, both inclusive, where n( = v + iv) is the num- 

 ber of trials, and v and w are proportional to the chances of arrival 

 or non-arrival in a single trial. That is, if the number of times 

 which A will happen in n trials be called A„, the preceding formula 

 is the probability that u, as deduced from 



A„ = np + u, [p = jj , 



will lie between - I and + ^ : on the supposition that p is given, and 

 A„ to be found by trial. And both Laplace and M. Poisson inune- 

 diately infer that the preceding result therefore represents tlie same 

 probability in the case where A^ has been observed, and p is to be 



* See the Addition at the end. 



