ON DIRECT PROBABILITIES. 39 



very small, out of the whole, and calculation would con- 

 firm it. 



Before laying down any more specific rules, I shall 

 take an instance of a somewhat complicated deduction, 

 in order to show that we do not really require any 

 other principle than is contained in the measure of 

 probability. Suppose A and B to play at whist against 

 C and D. A begins, holding ace, king, and queen of 

 trumps, and these trumps only : what right has he to ex- 

 pect that, hy playing them out in succession, he will force 

 all the trumps of the other party ? That is to say, ten 

 trumps being distributed among B, C, and D, and any 

 single card having the same chance of belonging to either 

 of the three, what is the probability that neither C nor 

 D holds more than three trumps ? 



Firstly, as to the number of tenures, which are per- 

 fectly distinct. To find the number of ways in which any 

 number of distinct objects can be divided among any 

 number of persons, use the following RULE : 



Multiply together numbers equal to the number of 

 persons as often as there are things to be divided among 

 them. Thus, to find in how many different ways ten 

 distinct cards can be divided among three persons, 

 find 



3x3x3, &c. (ten threes) or 3 10 which is 59049. 



The question now is, how many of these 59049 

 ways favour the supposition that neither C nor D holds 

 more than three out of ten. Both together they cannot 

 hold more than six : if, then, we pick out any given 

 six of the ten trumps, that set may be divided 

 among C and D in 2 6 or 64 ways. But of these, there 

 are two ways in which and 6 may be held, twelve 

 ways for 1 and 5, and thirty ways for 2 and 4, none 

 of which must be included. It must be observed, that 

 in the last sentence, the six distinct ways into which 

 6 may be divided into parcels of 1 and 5 must be 

 doubled, because * each gives two of our cases, ac- 



* It is important to observe that no duplication must take place on this 

 account, if the two numbers be the same; for instance, in dividing 6 into 

 parcels of 3. 



D 4 



