398 H. S. JENNINGS AND K. S. LASHLEY 



Given m numbers, forming pairs; from these n numbers are 

 drawn. What is the most probable number of entire pairs that 

 will be drawn? And what is, consequently, the most probable 

 number of entire pairs that will be left? 



This problem I did not find explicitly taken up in any of the 

 books on probabilities which I consulted. It may be attacked 

 directly in the following way : 



Suppose that the total number m is 20 (forming thus 10 pairs), 

 so that we have in the hat the series 1 to 10, twice repeated. Now, 

 if we draw out one number, obviously no pair will be obtained. 

 There are then 19 numbers left, and if we draw out one more, 

 there is one among the 19 that will, with the first one drawn, make 

 one pair. Thus the chance for getting one pair when two mem- 

 bers are drawn is in this special case 1/19, and in general, it is 



— That is, if we repeat the process of drawing 2 from 20 



m — 1. 



a great number of times, we shall get a pair in 1/19 (or -j 



of all cases, while we shall get no pair at all in the remainder, or 



(7n—2\ 

 = T ), of all the cases, 

 m — 1/ 



Now, consider the case where one more number is drawn, mak- 

 ing 3. Before the third one is drawn, there remain 18 (orm — 2). 

 Now, as we have already seen, a pair will have been obtained with 



the first two drawn in 1/19 ( = • ) of all cases; in these cases 



\ m — 1/ 



no additional pair can be obtained when the third is drawn. But 



in the cases where the first two drawn did not form a pair (that is, 



7n 



_ 9 



in 18/19, or , of all cases), there are two numbers out of the 



m — 1 



18 (or m — 2) remaining that, with the two already drawn, will 



form pairs. Thus there are now two chances out of 18 for getting 



a pair when the third number is drawn. But this is true only for 



(fj2, 2\ 

 or r) of all cases. So the total chance from this 

 772 — 1/ 



source for drawing a pair is 2/18 of -- ( or of ), which 



19 \ m — 2 m — 1/ 



