BIPARENTAL INHERITANCE IN PARAMECIUM 401 



For a formula which for even numbers drawn gives correctly 

 the most probable number of pairs I am indebted to Dr. A. B. 

 Coble, of the Mathematical Department of The Johns Hopkins 

 University. This formula is (if k represents the most probable 

 number of pairs) : 



(n + 1) (1+ l) 

 k = A| L (2) 



TO + 3 ^ ' 



In this case the nearest integer below the result gives the most 

 probable number of pairs (if the result is itself integral, then this 

 and the integer below it are equally probable). Thus, if we draw 

 10 from 20, the most probable number of pairs is given by 



^^nx6^ 20 

 23 23 



so that the most probable number of pairs is 2. But this formula 

 is not available when odd numbers are drawn. We shall see later 

 an indirect method of determining with absolute certainty the 

 most probable number of paii's when an odd number of units is 

 drawn. 



But it is often needful to know what is the relative probability 

 of a gi^^en number of pairs being di'awn, even though this may 

 not be the most probable number. For example, in Miss Cull's 

 case, cited above, we found that the most probable number of 

 entire pairs that will be included when 83 out of 186 die is 18, 

 while the actual number of pairs included is 28. What is the 

 probability that we should get 28, if the distribution of deaths 

 has no relation to the pairing? 



For a formula to determine the probability of any given num- 

 ber of pau's when a given number of units is drawn, I am again 

 indebted to Dr. A. B. Coble, to whom I wish to express my 

 thanks. If x represents the probability of any given number of 

 pairs k, then Coble's formula is as follows: 



n! m-n! Z!2°-2k 

 ^ ~ m\k\n-2k\l-n + k\ ^ 



