SOME BIOLOGICAL PROBLEMS, 12/ 



\in + 11 



Vt^/"(i -PT- (i) 



From (i) we can obtain directly the probability that exactly m 

 males and Ji females occur in in + n drawings and the expression 

 will be 



hn -\- 11 



P = n — ^ •*'( I — xY (2) 



^ \in\n ^ ■' ^ ^ 



where x denotes males and [i — x) females. 



If we call the right hand member of (2) j/, then 



y = cx'"\ I ~ ,t')" 



and this equation represents a curve with a zero ordinate when 

 X — o and ;r= i. Since the elementary area under the curve is 



\m -\- n 



I , ^"'(i —xYdx, 



\tn\n ^ ' 



the number of cases which occur between the two limits {a, b) is 

 proportional to the sum of the differentials taken from a to b, or 



\m + 



\in \n 



n r 



= I .r'"(i - xfdx. 



U a 



Also the totality of cases will be proportional to the sum of the 

 differentials taken from o to i, or 



;// + J I 



un \n 



I x"\i — x^dx . 



Hence the probability that the ratio of the males in the jar to 

 the entire lot hes between the two limits is 



I ;ir"'( I — x^dx 



/='4 (3) 



I x\\—xYdx 



Jo 



This is the well-known theorem of Bayes.' This equation as it 



1 See Todhunter's " History of the Theory of Probability from the Time of Pascal 

 to that of Laplace," 1865. 



