VOL. XXVII.] PHILOSOPHICAL TRANSACTIONS. 607 



that is, one chance for m double, one for f double, and 2 for m single and p 

 single; in four such dice there are chances m* + 4m^p + 6m^f^ + 4mp^ -|- p*; 

 that is, one chance for m quadruple, one for p quadruple, 4 for triple m and 

 single p, 4 for single m and triple p, and 6 for m double and p double: and 

 universally, if the number of dice be n, all their chances will be expressed in 

 this series, 



M" + ^ X M"- P + ^ X ^ X M'-'P^ + ^ X ^ X ^ X M'-» P^ -f &C. 



It appears plainly, that when the number of dice is even, there are as many 

 m's as p's in the middle term of this series, and in all the other terms there are 

 most m's or most p's. 



If therefore a man undertake, with an even number of dice, to throw as 

 many m's as p's, he has all the terms but the middle term against him; and his 

 lot is to the sum of all the chances, as the co-efficient of the middle term, is 

 to the power of 2 raised to an exponent equal to the number of dice : so in 

 2 dice, his lot is ^ or -^, in 3 dice -jV or f ; in 6 dice ^ or -rV; in 8 dice -^ 

 or tVf; &c. 



To find this middle term in any given power or number of dice, continue 



the series - X ^^—— X ^^-5— &c, till the number of terms be equal to -i-n. For 

 example, the co-efficient of the middle term of the J 0th power is y X 4 X -f 

 X -I- X X = 252, and the 10th power of 2 is 1024; if therefore a undertake 

 to throw, with 10 dice in one throw, an equal number of m's and p's, he has 

 252 chances out of 1024 for him, that is his lot is -tVtt or -5^ , which is less 

 than 4-. 



It will be easy by the help of logarithms, to extend this calculation to a very 

 great number, but that is not my present design. It is plain from what has 

 been said, that with a very great number of dice, a's lot would become very 

 small ; and consequently (supposing m to denote male and p female) that in the 

 vast number of mortals, there would be but a small part of all the possible 

 chances, for its happening at any assignable time, that an equal number of 

 males and females should be born. 



It is indeed to be confessed that this equality of males and females is not 

 mathematical but physical, which alters much the foregoing calculation; for in 

 this case the middle term will not exactly give a's chances, but his chances will 

 take in some of the terms next the middle one, and will lean to one side or the 

 other. But it is very improbable (if mere chance governed) that they would 

 ever reach as far as the extremities : but this event is happily prevented by the 

 wise economy of nature ; and to judge of the wisdom of the contrivance, we 

 must observe that the external accidents to which males are subject (who must 



