USE OP TABLES. 89 



50x2=100, VIo61=10 S2 ~ 18 _i-4o_ 

 H'=-15891 15891x50 = 7-9455 



^32x 18 = 24, 8 x 24 x 1-40=268-8 



7-9 divided by 268-8 is ^ nearly: 1 -^ is r ?|i 



Hence it is about 26l to 8 that A was more probable 

 than B. 



ADDITIONAL RULE. When a and b are nearly equal, 

 find t, as in the last rule; find H (not H') correspond- 

 ing to t, add I, and divide by 2 : the result is the pro- 

 bability required. 



The additional rule belongs to the more important 

 case of the two, namely, that in which A has not hap- 

 pened so much oftener than B as to justify an imme- 

 diate conclusion that it was the more probable event of 

 the two. Suppose, for instance, that A has occurred 

 10,100 times out of 20,000 trials, and B 9,900 times: 

 then t = 200 divided by 200, or 1 ; to which H is 

 843, and this increased by 1, and the result divided by 

 2, gives -922. It is, therefore, about 11 J to 1 that A 

 was the more probable. 



The preceding solution can be applied to various 

 species of observations ; of which we shall see more 

 hereafter. The following may be considered as closely 

 connected with it. If we make two different sets of 

 trials, in circumstances which we suppose to be the 

 same, it will generally happen that the As will not bear 

 the same proportion to the Bs in both sets. If r for 

 instance, we find 1000 As arrive in 2000 trials, the 

 odds are very much against the arrival of exactly 5000 

 As in a new set of 10,000 trials, though the expectation 

 is that something near that number of As will arrive. 

 Suppose that the first and second sets of trials give 1st, 

 50 As, 30 Bs; 2nd, 112 As, 6l Bs. 



In the second set the As bear a larger proportion to 

 the whole than in the first : and our present question is 

 what presumption thence arises that there is some dif- 

 ference of circumstances between the two sets, which 



