£0S O^ THE DOCTHINES OF CHANCE. 



not only to Opsimath, but to many other of your coriespon- 

 dents. 



Question. 



A, with a truths, tells b talsehoods, and B with c truths, d 

 falsehoods, what is the probability of the truth of a Circum- 

 stance in which they both agree ? 



Sola (ion. 

 Piobability of As they are supposed to agree in their relation, they must 



the truth of a 

 fact related by 



either both speak truth or both falsehood. Since then the pro- 

 two persons, bability of an event's happening is always expressed bv the 

 quotient of the number of times in which it may both happen 

 and fail, it is evident, that in the present case, the probability 

 of a circumstance being true, in which both of them agree, 

 will be expressed by the quotient of the number of times in 

 which they may agree in telling truth divided by the number 

 of times in which they may agree in telling both truth and 

 falsehood. Now the number of times in which they may 

 agree in telling truth will be the number of combinations of 

 a in c, viz. a c (for each of the truths a may be lulJ with each 

 of the truths c) ; and for the same reason tlie number of 

 times in which both of them may agree in telling falsehood, 

 will be 6 <i; the true expression tiicrefore for the probability 



required will be — [17;^. 



a c 

 (jry^ 1^ Cor. 1. Let a, b, c, and d be all equal, then will TTTv 



rr |, now the probability of A's telling truth is expressed by 



a 



■ ■, which will also = |; hence the probability of the 



truth of a circumstance in the relation of which two persons 

 agree who are each in the habit of relating truths as often as 

 lalsehood, will be the same as if related by either of them se- 

 parately. 

 Pyj. 2 Cor. 2. If a be greater than h and c greater than d, then 



a a c c a c 



— r-r ~ - — TT" and " ,--} :::: ; — iNow since a is greater 



than 



