56 ESSAY ON PROBABILITIES. 



probability that A was the state which produced the 

 event : and similarly for the rest. [Or, reduce the re- 

 sults of the first part of the rule to a common deno- 

 minator^ and use the numerators only in the second 

 part of the rule.] 



EXAMPLE I. There is a lottery which is one or 

 other of the two following : 



(3 white, 7 black) (all white). 



A ball is drawn, and restored; this takes place five 

 times, and the result is always a white ball. What are 

 the chances for each lottery ? 



Upon the supposition that the first lottery was that 

 in question, the chance of the observed event is the 

 product of -^ tV io> fo> and TO> or Ti >VbW When 

 the second is the lottery, the observed event is certain, 

 and its probability is 1 or -ViyaoSo- Consequently, the 

 probability for the second lottery is -JSJJ-g^-j, or the 

 second has the odds 100000 to 243, or more than 411 

 to 1 in its favour. 



EXAMPLE II. Two witnesses, on each of whom it 

 is 3 to 1 that he speaks truth, unite in affirming that 

 an event did happen, which of itself is equally likely 

 to have happened or not to have happened. What is 

 the probability that the event did happen ? 



The fact observed is the agreement of the two wit- 

 nesses in asserting the event: the two possible ante- 

 cedents (equally likely) are, 1. The event did happen. 

 2. It did not happen. If it did happen, the probability 

 that both witnesses should state its happening is that 

 of their both telling the truth, which is f x J, or ^. 

 If it did not happen, then the probability that both 

 witnesses should assert its happening is that of their 

 both speaking falsely, which is x J, or -*$. Conse- 

 quently, the probability that the event did happen is 

 the (9 -f l)th part of 9, or T ; that is, it is 9 to 1 

 in favour of the event having happened. 



EXAMPLE III. There are two urns, having certainly 

 3 and 2 white balls; and in one or other, but which 



