A MATHEMATICAL TREATMENT OF SOME 

 BIOLOGICAL PROBLEMS. 



SHINKISHI HATAI, 

 Associate in Neurology, The Wistar Institute of Anatomy. 



Dr. King's work ('07, '09), "On Studies on Sex-determination 

 in Amphibians," suggests an interesting problem which may be 

 put in the following form : 



A jar contains a large number of male and female tadpoles, the 

 proportion of each being unknown : if on picking out in ■\- n tad- 

 poles, in are found to be pnales and n females, to find the prob- 

 ability that the ratio of the number of either sex to the entire lot 

 lies between given limits. 



It will be seen that problems of this nature occur frequently 

 in biological investigations and that it is of importance to have 

 some method for determining the accuracy of the observed pro- 

 portions. This can be done by means of the formula given below. 

 As the development of the formula is somewhat complicated I 

 shall present the entire process of the mathematical treatment of 

 the solutions based on the theorem of Bayes, together with one 

 application. Although such an elementary exposition of the sub- 

 ject will be superfluous for one who is familiar with the theory of 

 probabilities nevertheless for others it may be helpful. 



If p denote the probability that an event will happen, then 

 (i — />) is the probability that the event will fail. If the proba- 

 bility that the event will fail on any single trial is (i — /), the 

 probability that it will fail every time is (i — /)". The proba- 

 bility that it will happen on the first trial and fail on the succeed- 

 ing n — \ trial is/(l — i>)"~^ But the event is just as likely to 

 happen on the second, third, etc., trials as on the first. Hence 

 the probability that the event will happen just once in the n 

 trials is 



np{i-pr-\ 



Continuing this process, we can easily see that the probability 

 that it will happen ni times in m + n trials is 



126 



t 



