7* ESSAY ON PROBABILITIES. 



whole result of 3000 trials, we shall have drawn the 

 three in very unequal numbers ; so that, destroying the 

 distinction between A and A' 3 we feel secure of drawing 

 A twice as often as B ; and it is obviously two to one 

 in favour of A at each trial. The following phrases 

 seem to common sense to mean the same thing. 



It is two to one that A 

 shall happen, and not B. 



It is an even chance for 

 head or tail. 



It is more than a hundred 

 to one, that a ship at sea will 

 not be lost. 



In the long run, A will 

 happen twice as often as B. 



In a large number of tosses, 

 the heads and tails will occur 

 in nearly equal numbers. 



Of all the ships which sail, 

 the number which is not lost 

 exceeds that which is lost more 

 than a hundred times. 



I now proceed to some problems, which will exhibit 

 the method of applying the tables, and will illustrate 

 and confirm the preceding notions. 



PROBLEM I. The odds for A against B being a to b, 

 to find the chance that in n times a -f- b trials, A shall 

 happen exactly n X a times, and B n X b times. 



RULE. Divide the II' belonging to t = (page 72) 

 by the square root of the following : 8 times the product 

 of n, a, and 6, divided by a + b. 



Suppose, for instance, a die is thrown 6000 times ; 

 what is the chance that exactly 1000 of the throws shall 

 give an ace ? Here it is 1 to 5 that an ace shall be 

 thrown in any one trial, and 6000 is 1000 times 1-f 5. 

 Hence a = 1, b = 5, n = 1000 : 8 times the product 

 of n, a, and b is 40,000, the sixth part of which is 6667 

 (sufficiently near), and the square root of this is 81*65. 

 Again, when.* = 0, we have in Table I. 



A = 112833, A2 = H ; whence H' is 1-12844 



and 1-12814 divided by 81-65 gives -014 very nearly. 

 This is very near the real probability that 6000 throws 

 with a die shall give exactly 1000 aces: for such an 

 event there are only 14 chances out of a thousand ; and 



