of the Hypergeometrical Series, 245 



greater than 10, and the whole distribution might be closely 

 represented by drawing 10 balls 18,600 times out of a bag 

 containing 17 white and 48 black balls, and counting the 

 white occurrences in each draw. Actually the frequency was 

 obtained by drawing 10 cards out of an ordinary pack of 52 

 and counting the hearts in each draw. Thus we have : — ■ 



Actually. From theory. 



Start at -'0244 



Mean 25 2 4988 



Number drawn .. . 10 10-1055 



Base unit 1 '9721 



p -75 -7432 



q '25 -2568 



n 52 65-2034 



Now it is clear that the first six results are in good agree- 

 ment, but that n diverges from its actual value by 25 per cent, 

 although the number of trials, 18,600, is far larger than are 

 recorded in most practical cases. 



It is of interest to record the actual and theoretical 

 frequencies : — 



No. of hearts. Observed. Theory. 



■0 ...... 759 747-5 



1 3277 3239 



2 5607 5642 



3 .. .. 5159 5172 



4 ...... 2701 2743 



5 907 87L 



6 165 166 



7 24 18-5 



8 1 1 



y o o 



10 : o o 



The deviations are four positive and four negative, and 

 four above and four below their respective probable errors. 

 Thus the experimental results are in good accordance with 

 theory. 



Notwithstanding this, n has a large deviation from its 

 theoretical value when determined by moments. It is clearly 

 a quantity, when thus determined, liable to very large probable 

 error. Thu«, while the problem is theoretically fully solved — 

 ana 1 it is difficult to believe that any other solution can have 

 less probable error — yet we meet, unless we take an immense 

 number of trials, with large variations in our estimate of the 

 number from which the drawing is made. I have tested this 



Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. S 



