404 H. S. JENNINGS AND K. S. LASHLEY 



Since the sum of all the probabihties is 1, it is practically only- 

 necessary to find the probabilities for all numbers showing less 

 than the given deviation (in this case, for all numbers from 19 to 

 37) and subtract their sum from 1 ; the remainder will be the sum 

 of the probabilities for all deviations as great as (or greater than) 

 the given deviation. This procedure is that which in the Appen- 

 dix is formulated as rule (5). 



For finding the probabilities for a series of successive numbers 

 (as 19 to 37, in the case above), the following formula (4) will be 

 found convenient. First find the probability by formula (3) 

 for the lowest number of pairs (in this case 19). Call the proba- 

 bility for this number Xi. Then the probability for the next 

 higher number of pairs (which we may call ko; in this case ^2 is 20) 

 is given by multiplying the probability Xi by 



(n-2 h+1) (n-2A-2+2) . 



4 ^2 (1 - n + /C2) 



The probability for the next higher number of pairs (in this 

 case 21) may then be found from this result by using the same 

 formula anew, and so on for the entire series of numbers. (Or 

 if we desire we may begin with the higher numbers, and find the 

 probability for succeeding lower numbers by using the expres- 

 sion (4) inverted.) 



An example may be taken. In the case we have cited, we 



find that the probability for 19 pairs is, by formula (3), 0.000059733. 



To find the probability for 20 pairs we must multiply this value, 



64 X 65 



according to formula (4), by :, —r, which gives 0.00031061 



& ^ ^' -^ 4 X 20 X 10 



as the probability for 20 pairs. To find the probability for 21 



pairs we must now multiply this value by - — ^^ — :, giving 



0.001313 as the probabihty for 21 pairs. 



Proceeding in this way, and adding together the ^'alue of the 

 probabilities obtained, we find that the total probability for all 

 numbers from 19 to 37 is 0.99995242; this then is the probability 

 that the deviation shall be less than the observed deviation 10, 

 if the distribution of deaths has no relation to the pairing. The 



