26G JOURNAL OF THE WASHINGTON ACADEMY OF SCIEXCES VOL. 11, NO. 12 



teriapercc, we have B/N = aX/n. Hence equation (2) reduces to 



P^ = exp ( — aX) * 



^^ C! ^ (3) 



which is the fundamental expression for the probability Pc of finding 

 C bacteria in an a cc. sample of water containing X bacteria per cc. 

 The sample usually taken is a = 1 cc. From the form of the expres- 

 sion we have 



00 



^C{Pc)=l (4) 







as it should, to represent probability. The probability Pq of a neg- 

 ative result, on plating out, is 



Po = exp(-aX) (5) 



and of a positive result is 1— Po- This reduces to McCrady's- result 

 when aX = 1, namely 1 — Po = 1 — l/e = 0.63. 



As there has been some misunderstanding of McCrady's work, per- 

 haps due partly to the form in which it was expressed, a few remarks 

 on the theory here given may be appropriate. The only assumptions 

 are: (1) that in a large number of samples of a cc. each, the bacterium 

 considered is on the whole as often in one of the samples as in another 

 (random distribution), and (2) that the presence of one bacterium in 

 a sample does not affect the chances of the others being there (inde- 

 pendent probabilities). These assumptions can hardly be doubted in 

 this particular case, because one bacterium occupies about 10~^^ cc, 

 so that one million per cc. would occupy only one millionth of the vol- 

 ume. From studies of Brownian particles it is known that there are 

 considerable fluctuations in density. 



Now suppose the samples are diluted, /3 cc. in D cc. of water, then 

 the probability of finding C bacteria in a cc. of the diluted sample is, 

 from (3) 



where 



zC 

 Pc=~exp.{-z) (6) 



X 



a|3 — 

 D 



2H. H. McCrady. Journ. Infectious Diseases 17: 183. 1915. 



