AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 405 



by 1 + 2e, + SeiBa + 2e,e2 + ... + 2e,ea ...e,„, the I being omitted if all cannot be false. The 

 various restrictions which might be imposed, as that only an even number can be true, that no two, 

 three, or any number of contiguous cases can be true together, &c. &c. may be easily contained under 

 this one rule. In every set of cases that can be true together, multiply together the exponents of 

 those cases ; the product is the numerator of the probability that those cases only are true, and 

 the sum of all the products is the denominator. 



This rule applies to one case which we have not yet considered. When several arguments were 

 proposed together, all for, or all against, a conclusion, it was supposed that they were perfectly 

 independent. But it may hap[)en that two or more arguments are so connected that some must be 

 valid together and invalid togetlier, or that some are valid when others are invalid, and vice versa, 

 or that the validity of one makes another valid, but the invalidity of the first has no influence on the 

 validity of the second. All these cases, and a great number of otiiers, including in fact, under one 

 view or another, any question that may be proposed, mav all be solved by the following Rule. 



There is any number of events, each of which may happen in any nun)ber of ways, the separate 

 probabilities of which are given, but so connected tiiat there are specific necessary coincidences, 

 or failures of coincidence. Take all the combinations which can happen, and compute the 

 probability of each combination, as if its events were entirely unconnected. The resulting products 

 are proportional to the probabilities of the several cases arising. 



Thus, if there were three urns, the first giving white, black, or red (with chances tv, b, r) ; 

 the second white or black (with chances w',//) ; the third while or black (with chances w'', b"), but so 

 connected that black cannot be drawn from tiie first, nor white from all three, nor red from the first 

 excejJt when different colours come frou) the second and third, and it be required to find the chance 

 of having a red ball, we proceed thus. Enumerate the pos.sible cases, which are WWB, WBW, 

 WBB, RBW, RWB, and the probability of a red ball is 



)• (b' w" + w' b') 

 w(w'b" + b'w" + b'b") + r{b'w" + w'b")' 



I have taken such an example, because it seems as if the condition that a black ball cannot be 



drawn from the first is equivalent to taking away those black balls, in which case the chances 



of the others cannot be w and r. But if the black balls be previously removed, then for w and r 



w r 



we must write and , which will not affect the formula. In the same way any addition 



1-61-6 •' •^ 



of other coloured balls, with the condition that they cannot be drawn, though it will affect the 



probabilities of the independent events made use of in the solution of the problem, will not affect 



the ratio which expresses the final result. 



I have given so many proofs of particular eases of this principle that it is not necessary 

 to say any thing on the general proof. But I shall observe that the circumstance noticed in 

 combining argument and testimony, namely, that instead of the validity of an argument entering _/«)• 

 the conclusion, the invalidity enters against, — is an immediate application of the preceding rule. 

 For it is not the validity of an argument which is necessary to the truth of a conclusion, but the 

 invalidity of it which is necessary to its falsehood. Thus, in Problem 4, the necessary cases are, cither 

 1. Argument against invalid, and testimony for true, giving (1 - 6)/i; or 2. Argument for invalid, 

 and testimony against true, giving (1 - a) (l -/u). 



The application of the principles on which the preceding rule is established, would, I suspect, 

 give much clearer views of many problems than the ordinary method of employing inverse con- 

 Kiderations. 



A. DE MORGAN. 



llniversily College, London, 

 October 3, 1846. 



