76 ESSAY ON PROBABILITIES. 



number I : I being small, compared with the whole number 

 of As in the probable mean. 



RULE. Divide twice / by the square root obtained in 

 the last example ; and find the value of H'from Table I., 

 taking the preceding quotient for t. DivideH', so found, 

 -by the square root just used, and the quotient is the 

 answer required. 



N. B. This rule also applies when the number of A s 

 is to exceed the probable mean by /. 



EXAMPLE I. In 6000 throws with a die, what is 

 the chance that the aces shall fall short of (or exceed) 

 1000 by exactly 50 ? Here a = 1, b = 5, n = 1 000, 

 and the square root is 81*65, as before. And twice /, 

 or 100, divided by 81'65, gives 1-22: to which the 

 value of H' is 25162 -f- | of 6ll, with five decimal 

 places, or '25468. This last, divided by 81 '65, gives 

 0031 ; so that it is about 997 to 3, or 332 to 1, against 

 the proposed event : and the 6000 throws must be re- 

 peated 232 times to give an even chance of succeeding 

 once. 



EXAMPLE II. What is the chance that in 200 tosses, 

 there shall be exactly 95 heads ? Here a = 1, b = 1, 

 n = 100, I =. 5, and the square root, as before, is 20. 

 And twice I, or 10, divided by 20, gives "50, which 

 being t, the value of H' is '87882, which divided by 

 20 gives -044 very nearly. It is, then, 956 to 44, or 

 about 22 to 1, against the proposed event. 



EXAMPLE III. In 12 tosses, what is the chance of 

 exactly 7 heads ? Here a = 1, 6=1, n = 6, /=- 1, 

 the square root, as before, is 4*9., and 2 divided by 4'9 

 is '41 nearly ; which being t, H' is '95384, which 

 divided by 4*9 gives -194. It is therefore 806 to 194, 

 or 4-,-^j- to 1, against the proposed event. In page 47- 

 it is 3304 to 792 against this event, or 4^! nearly. Hence 

 the incorrectness of our rule is very small. 



PROBLEM III. The odds for A against B being 

 a to b y required the chance that in n times a -f b throws, 

 the number of As shall not differ from the probable 

 mean by more than /. 



