402 H. S. JENNINGS AND K. S. LASHLEY 



where 



m = the total number of individuals 

 I = the total number of pairs (so that I = ^ m) 

 n = the number of individuals drawn 

 k = the number of pairs whose probability we desire 



n\ = the product of all integers up to n (thus 186! is the product of all integers 

 from 1 to 186) 



This formula forms the basis for further formulae which I 

 have developed from it (notably number (4), given below), 

 and it is indeed the basic formula for the greater part of our work. 

 It may be employed to determine directly either the probability^ 

 of any given number of pairs among those drawn, or of any given 

 number of pairs among those not drawn. 



Tn a concrete case the formula works out as follows: Suppose 

 that out of 20 individuals (10 pairs), 9 individuals are drawn. 

 What is the probability that there will be just 3 pairs included 

 among those drawn? 



Here w = 20; n = 9; /b = 3. The formula (3) therefore be- 

 comes 



^ 9!11!10!23 

 ^ ~ 20!3!3!4! 



which gives x = 0.2005, so that there is almost exactly one chance 

 in five of getting 3 pairs when 9 are drawn from 20. We may put 

 the question in the reverse way by asking: What is the probabil- 

 ity of there being left 4 pairs when 11 are left out of 20? (If 

 9 are drawn from 20, and 6 of the 9 from 3 pairs, then the other 

 3 belong to other pairs, so that 6 pairs are represented among 

 those drawn, leaving 4 complete pairs among those not drawn). 

 In this case m is 20, n is 11, while k is 4, and formula (3) there- 

 fore becomes 



^ 1119! 10! 2« 

 '^" ~ 20T4r3!3! 



which is identical with the formula (given above) for 3 pairs when 

 9 are drawn from 20. Thus, the probability of the number of 

 pairs actually drawn is of course bound to be the same as that for 

 the number of pairs actually left, as was previously mentioned. 



