USE OF TABLES. 75 



it is 1000 14 to 14, or about 70i to 1, against the 

 event. This result is rather above what we should have 

 expected ; we might have imagined it to be more 

 than 71 to 1 against 6000 throws giving exactly 1000 

 aces. 



As another example, let us find the probability that, 

 out of 200 tosses with a halfpenny, there shall be ex- 

 actly 100 heads and 100 tails. Here a = 1, b = 1, 

 n = 100, and 8nab* is 800, which divided by a + b, 

 or 2, gives 400, the square root of which is 20. And 

 IT, when t (or 1-12844) divided by 20, gives '056. 

 It is therefore about 944 to 56', or 17 to 1, against the 

 proposed event ; and (page 42.) we must repeat 200 

 throws 1 2 times to have an even chance of the equality 

 of heads and tails happening once. 



Generally speaking, the rules in this chapter are very 

 accurate only when the number of trials is considerable. 

 Suppose only 12 tosses; required the chances of 6 

 heads and 6 tails. Here o = 1, 6=1, n = 6, 

 Snab = 48, which divided by 2 gives 24, whose square 

 root is 4'9 very nearly. And 1*12844 divided by 4*9 

 gives -23, or 77 to 23, that is 3^ to 1 against the 

 event. That is (page 47)> this rule is not very inaccu- 

 rate, even when the number of trials is as low as 12. 



We shall call the event whose chance is sought in the 

 preceding problem, the probable mean; understanding 

 by that term the event which is more likely to happen 

 than any other. Thus, when 12 halfpence are thrown 

 up, 6 heads and 6 tails is the probable mean, being the 

 event which is more likely than any other, though not 

 in itself more likely than not. When 6000 throws are 

 made with a die, the probable mean is 1000 aces, 1000 

 deuces, &c. 



PROBLEM II. The odds for A against B being a to &, 

 required the chance that in n times a + b trials, the 

 As shall fall short of the probable mean by a given 



* Juxtaposition of numbers, in algebra, stands for their product. 



