100 ESSAY ON PROBABILITIES. 



494, at the second ; in 232, at the third ; in 1 37, at 

 the fourth ; in 56, at the fifth ; in 29, at the sixth ; in 

 25, at the seventh ; in 8, at the eighth ; and in 6, at 

 the ninth. Let us, then, compute the amount which he 

 would have received if he had bond fide played all these 

 games on the preceding terms. 

 1061 x 2 = 2122 



494 x 4 = 1976 



232 x 8 = 1856 



137 x 16 = 2192 



56 x 32 = 1792 



29 x 64 = 1856 



25 x 128 = 3200 



8 x 256 = 2048 



6 x 512 = 3072 



The 2048 games would have 

 given 20,114Z. or nearly 

 10Z. per game, one game 

 with another. 



No person would stake at this game for a single trial, 

 upon the prospect of head being deferred till the ninth 

 throw. Nevertheless, in this instance, it appeared that 

 out of 2048 trials, such a rare occurrence happened often 

 enough to realise more than any other, with one excep- 

 tion. If Buffon had tried a thousand times as many 

 games, the results would not only have given more, but 

 more per game. A larger net would have caught, not 

 only more fish_, but more varieties of fish ; and in two 

 millions of sets, we might have expected to have seen 

 cases in which head did not appear till the twentieth 

 throw. Let us turn back to page 43, and inquire, by 

 the rule there given, in how many trials it is 10,000 to 

 1 that head will be deferred till the twentieth throw. 

 Out of 2 20 , or 1,048,576 cases, representing the number 

 of different arrangements which may happen in 20 

 throws, the arrangement in question is but one ; it is 

 then 1,048,575 to 1 against its arrival in any one given 

 trial. Look in the Table opposite to " 10,000 to 1," 

 and we find 921: multiply 1,048,575 by 921, and 

 divide by 100, which gives 9)657,375. It is then 

 more than 10,000 to 1 that head is deferred till the 

 twentieth throw somewhere in ten millions of trials, and 

 more than an even chance that it is found to occur in 

 seventy thousand trials. Thus the reader may readily 



