GAMING. 



probability of the first happening be- 

 fore the second is 2, but the probability 

 of its happening twice before it, is but 

 ^ X -| or * ; therefore it is 5 to 4 se- 

 ven does not come up twice before four 

 once. 



But if it were demanded what must 

 be the proportion of the facilities of the 

 coming up of two events, to make that 

 which has the most chances come up 

 twice, before the other comes up once : 

 The answer is 12 to 5 very nearly; whence 

 it follows, that the probability of throw- 

 ing the first before the second is ^.l, and 

 the probability of throwing it twice is * ^L 

 X jf, or .|4| . therefore the probability 

 of not doing it is ^||: therefore the odds 

 against it are, as 145 to 144, which comes 

 very near an equality. 



Suppose there is a heap of thirteen 

 cards of one colour, and another heap of 

 thirteen cards of another colour, what is 

 the probability, that, taking one card at a 

 venture out of each heap, I shall take out 

 the two aces ? 



The probability of taking the ace out 

 of the first heap is -l^, the probability of 

 taking the ace out of the second heap is 

 _?_.; therefore the probability of taking 

 out both aces is J^ X ^ => T j^> which 

 being subtracted from 1, there will re- 

 main \i.||: therefore the odds against me 

 are 168 to 1. 



In cases where the events depend on 

 one another, the manner of arguing is 

 somewhat altered. Thus, suppose that 

 out of one single heap of thirteen cards 

 of one colour, I should undertake to take 

 out first the ace ; and, secondly, the two : 

 though the probability of taking out the 

 ace be _1_ , and the probability of taking 

 out the two be likewise T * ; yet the ace 

 being supposed as taken out already, 

 there will remain only twelve cards in 

 the heap, which will make the probabili- 

 ty of taking out the two to be __; there- 

 fore the probability of taking out the ace, 

 and then the two, will be J^ X -jL- 



In this last question the two events 



have a dependence on each other, which 

 consists in this, that one of the events be- 

 ing supposed as having happened, the 

 probability of the other's happening is 

 thereby altered. But the case is not so 

 in the two heaps o*f cards. 



If the events in question be n in num- 

 ber, and be such as have the same num- 

 ber a of chances by which they may hap- 

 pen, and likewise the same number b of 

 chances by which they may fail, raise 

 a _j- b to the power n. And if A and B 

 play together, on condition that if either 

 one or more of the events in question hap- 

 pen, A shall win, and B lose, the probabili- 



ty of A's winning will bp a "J"^'-JH_- ; and 



n 



that of B's winning will be 



; for 



when a -j- b is actually raised to the 

 power n, the only term in which a does 

 not occur is the last b n ; therefore all 

 the terms but the last are favourable 

 to A. 



Thus, if n = 3, raising a + b to the 

 cube a3 -f 3 a- b. -f 3 a 6 a -f te, all the 

 terms but to will be favourable to A; 

 and therefore the probability of A's 



. . ... , o3 + 3 tfb+'Sab* 

 winning will be - _ ^- -- , or 



_ 

 0+6)3 



-L ; and the probability of B's 



winning will be 





But if A and B 



play on condition that, if either two or 

 more of the events in question happen, A 

 shall win; but in case one only happen, or 

 none, B shall win; the probability of A's 



, a-f- bin n a A" 1 bn> 



winning will be === 



n + b\n 



for the only two terms in which a a does 

 not occur are the two last, viz. nab ni 

 and b n . See Simpson's "Nature and 

 Laws of Chance." We shall now add a 

 table that may be useful to persons not 

 skilled in mathematics, and which is ap- 

 plicable to many subjects : 



