856 CONDORCET. 



Thus we have the following result in the Theory of Probability : 

 if the probability of a correct decision is the same for every voter 

 and is greater than the probability of an incorrect decision, then 

 the probability that the decision of the majority will be correct 

 becomes indefinitely nearly equal to unity by sufficiently in- 

 creasinof the number of voters. 



It need hardly be observed that practically the hypotheses on 

 which the preceding conclusion rests cannot be realised, so that 

 the result has very little value. Some important remarks on the 

 subject will be found in Mill's Logic, 1862, Vol. II. pages Qo, Q>Q, 

 where he speaks of '' misapplications of the calculus of probabilities 

 which have made it the real opprobrium of mathematics." 



666. We again return to the equation (2) of Art. 663. 



If we denote by -v/r (^q) the probability that there will be a 

 majority in favour of an incorrect decision, we can obtain the 

 value of yfr^q) from that of ^ (q) by interchanging e and v. 



We have also ^ (^) + '*/^ fe) = 1. 



Of course if v = e we have obviously ^(^q) ='\fr {q), for all 

 values of q ; the truth of this result when q is infinite is esta- 

 blished by Condorcet in a curious way ; see his page 10. 



667. We have hitherto spoken of the probability that the 

 decision will be correct, that is we have supposed that the result 

 of the voting is not yet known. 



But now suppose we know that a decision has been given and 

 that m voters voted for that decision and n against it, so that m 

 is greater than n. We ask, what is the probability that the de- 

 cision is correct ? Condorcet says briefly that the number of com- 

 binations in favour of the truth is expressed by 



12^ + 1 



v'^e^ 



and the number in favour of error by 



12^ + 1 



e'\'". 



m n 



Thus the probabilities of the correctness and incorrectness of the 

 decision are respectively 



