92 ESSAY ON PROBABILITIES. 



The following is the rule pointed out by the theory of 

 probabilities: The expectation of fluctuation should 

 be greater to a person who proposes to try q new in- 

 stances, upon the assumption that p preceding instances 

 have fairly represented the long run, than it should be 

 to another person, who knows in what proportions the 

 As and Bs really exist ; and greater in the proportion of 

 the square root of p augmented by q to the square root 

 of p. Thus, if in the preceding case, John and Thomas 

 propose to embark in a matter which depends on 300 

 more trials, the proportion of the square root of 100 -f 300 

 to that of 100, being that of 2 to 1, it follows that, 

 whatever reason John may have to guard against the 

 possibility of 300 drawings giving x more than 150 As, 

 Thomas has as much reason to guard against 2 x more 

 than the same number. 



PBOBLEM (to be compared with that in page 77- ) 

 When a + b trials have happened to give a As and b 

 Bs, required the chance that in n times a -j- b new. 

 throws, the number of As shall not differ from na by 

 more than /. 



RULE. Divide one more than twice I by a square 

 root to be immediately mentioned, and the quotient being 

 made t, the value of H in Table I. is the probability re- 

 quired. 



The square root in the 

 former problem was that of the 

 product of 8, n t a, and b 

 divided by a + b. 



The square root in the 

 present problem, is that of 

 the product of 8, w, n+l t a t 

 and b divided by a -f- b. 



The additional rule in page 81. may also be applied 

 verbatim, the square root *now meaning the second 

 square root above given ; and the inverse rule (p. 82) 

 may be applied in exactly the same way. 



EXAMPLE. In 600 drawings A occurred 100 times, 

 and B 500 times ; what presumption thence arises that 

 in 6000 more drawings A would occur somewhere be- 

 tween 1000 50, and 1000 + 50, or 950 and 1050 

 inclusive ? (See page 77 for the corresponding pfo- 

 blem.) 



