40 ESSAY ON PROBABILITIES* 



cording as C holds 1 and D holds 5, or C holds 5 and 

 D holds 1 : and so of the other cases. Consequently, 

 out of a given set of six, there are 64 all hut 44, that 

 is, 20 ways, in which, when they happen, A could force 

 all the adversary's trumps. But every set of six yields 

 20 such cases : hence the ways in which C and D can 

 together hold six trumps, are 20 times C (6,10 j or 

 20 X 210, or 4200. Similarly, a given set of 5 trumps 

 may be held by C and D in 2 5 or 32 ways, of which 2 

 and 10 must be excluded. Hence 20 x C ( 5,10 } 

 or 20 x 252, or 5040 sets are the number favourable 

 to the event in this case. Four trumps can be held 

 in 2 4 or 16 ways, of which 2 must be excluded, or 14 

 X C {4,10] that is, 2940, is the number of favour- 

 able cases. On the supposition that C and D together 

 hold only 3, or 2, or 1 trump, no exclusions are neces- 

 sary, and the number of cases are 2 3 x C {3,10J 

 or 960, and 2 2 X C { 2,10 j or 180, and 2 X 

 C [l,!0] or 20; and there is one case in which C 

 and D hold no trumps. All the favourable cases are, 

 therefore, in number 



4200 + 5040 + 2940 + 960+180+20+1 or 13341. 

 The chance of A being able to force all the adversary's 

 trumps is -j-gjJ-J-g, or nearly 3 1 / to 1 against it. 



Given a fraction less than unity, and which has high 

 numbers in its terms, required a set of fractions which 

 shall be very nearly equal to it, and each of which shall 

 be nearer than any other fraction of the same order 

 of simplicity. Required, also, a near estimate of the 

 error committed in each case. 



RULE. First perform the process for finding the 

 greatest common measure of the numerator and deno- 

 minator. 



