30 STATISTICS OF SEX 



Then by the principles of the theories of probabilities, if a couple be 

 taken at random from the whole mass, the respective combined probabili- 

 ties that the couple will be of one of the classes, and the child a male, 

 will be: 



In class A, y->h(p-\-a). 



B, Vp. (2) 



C, %*(p a). 



of which the sum is p, as it should be. 



The problem before us is to find a criterion for deciding whether 

 the quantity a, which we may consider as the unisexual factor, and which 

 we shall call the coefficient of unisexuality , is or is not of appreciable 

 magnitude. Such a criterion is afforded by a count of males and females 

 in families of two or more children. The theory requires that, in a 

 family of a given number of children, we express the probable respective 

 numbers of males and females in terms of the factor a. 



The problem now assumes the following form : A parent couple, 

 taken at random from the whole mass, has n children; what is the 

 probability that s of these children will be males and n--s females ? 



Using the notation 



rn~\ _ n(n 1) (n 2) . . . (n s + 1) 

 L s] ~ 1 .2.3.. .s 



we have the well-known theorem that, if the probability of an event on 

 a single trial is p, the probability of its occurring s times on n trials is 



Putting for ^ the three values of the probabilities given in ( 1 ) we find 

 that the probabilities in question are : 



For class A, [ n ~\ (p + a) s (1 p a)"- 8 



For class B, 1"-]^' (lpY'* (3) 



For class C, 1 Q; )' (1 /> + a)"~ s 



Multiplying these expressions, as in (2), by the respective factors 

 1/2 /*>, li' and ~yj l > putting for brevity 



n s = r 



