OX ESSAY ON PROBABILITIES. 



By precisely the same reasoning, if there had been 

 1000 balls in the lottery, and if 157 had been drawn 

 white, the probability that 27 more drawings would 

 have given white balls, would have been 



S 1000^4 



(157 + 27 = 184) 100027 S 10001*7 



The difficulty of calculating S 1000 184 is insuperable: 

 but a mathematical theorem which we shall proceed to 

 explain, makes it very easy to find a near approxima- 

 tion to the preceding result, and the nearer the greater 

 the numbers in question. 



Take the sums of the powers of the different num- 

 bers, as follows : 



First powers 1+2+ 3+ 4 + 5+ 6 + 



Squares 1+4+ 9+ 16+ 25+ 36 + 



Cubes 1+ 8+ 27+ 64+ 125+ 216 + 



Fourth powers 1 + 16+ 81+ 256+ 625 + 1296 + 



Fifth powers 1 +32+243 + 1024 + 3125 + 7776 + 



and examine the sum of any number of terms in any 

 line, as compared with the term immediately below the 

 last in the sum j thus : 



1+2 + 3 + 4 1 + 16 + 81+256 + 625 



16 3125 



Form fractions with such sums as numerators, and their 

 compared terms as denominators, and observe how 

 much each fraction, so formed, differs from the fraction 

 written in the last column, as foUows : 

 ]+2_3_l 1 l+2+3_6_l 1 



~4 4~2 4 9 9~ 2 6 



ilj^l+i J^li * , and so on, 

 16 16 2 8 25 25 2 10 



whence 



S (any number) JL __ _J 



square of that number 2 twice that number 



Hence it follows, that when the number is large, the 

 preceding fraction is very nearly one half, or 1 -f- 2 -j- 



3 -f up to a large number, is very nearly one half 



the square of that number. 



