SECT. 20.] Credibility of Extraordinary Stories. 427 



out of 100,000 drawings (the number required in order to 

 obtain a complete cycle of all possible occurrences, as well as 

 of all possible reports about them) the two witnesses agree 

 in a correct report of the appearance of white in 81, and 

 agree in a wrong report of it in 999. The Probability there 

 fore of the story when so attested is yfj^ 5 tne ^ act therefore 

 of two such witnesses of equal veracity having concurred 

 makes the report nearly 9 times as likely as when it rested 

 upon the authority of only one of them 1 . 



20. When however the witnesses have many ways of 

 going wrong, the fact of their agreeing makes the report far 

 more likely to be true. For instance, in the case of the 1000 

 numbered balls, it is very unlikely that when they both mis 

 take the number they should (without collusion) happen to 

 make the same misstatement. Whereas, in the last case, 

 every combined misstatement necessarily led them both to 

 the assertion that the event in question had happened, we 

 should now find that only once in 999 x 999 times would 

 they both be led to assert that some given number (say, as 

 before, 25) had been drawn. The odds in favour of the 



1 It is on this principle that the right and m are wrong). And the 



remarkable conclusion mentioned on chance of its being rightly asserted 



p. 405 is based. Suppose an event &spy m (l-y) n . Therefore the chance 



whose probability is p ; and that, of that when we have an assertion 



a number of witnesses of the same before us it is a true one is 



veracity (?/), m assert that it hap- py m (l-y) n 



pened, and n deny this. Generaliz- py m (i~jj) + (1 -p) y n (l- y) m 



ing the arithmetical reasoning given 



- ,, which is equal to 

 above we see that the chance of the 



event being asserted varies as 



(viz. as the chance that the event But this last expression represents 



happens, and that m are right and n the probability of an assertion which 



are wrong; plus the chance that it is unanimously supported by m-n 



does not happen, and that n are such witnesses. 



