B ON PROBABILITY, 
Let A, B, C, D be the hands dealt to each player out of 48 cards, 
: a, b, ¢, d, the four honours. 
It is evident that A, B, C, D may be combined with «, b, ¢, d in as 
many different ways as a, b, ¢, d can be permuted, thatis,in 4 Xx 3 XxX 2 xX 1, 
or 24 different ways ; therefore the probability that 
45 47 3X 46... . . Bn 2 x1’ 
(12.¢ 1! 16... is +. 3. X 2.0.1) 
o Sin — EE esteem tees eee. rr 24. 
each hand has an honour CE EE BE xX 2 
(13 X 123 JJ LA, VG 0 B.% 1} 
134 Quds againsty 
19% out 
=X 24 oon 17:2 
52 xX 51 x b0 x 49 20825 
So it may be found that the probability that 
Li hand has the oe ao 11x10. , _. 220 Ses] 
honours . « . [752x51%50x49 20825’ : 
s y 13x12x13 x12 2608 
Re hands have a ea Ws: 2 
two honours . . [T52x51 x50 x49 20825 
: 13x13 x13%12 12168 
one hand has two ho-]__ 13X13 X13 % st 125 128 6 2 
noursandtwohandsone| 752 x 51 x 50 x 49 20825 
one hand has three ho-] __ 13X13 x 12x11 x12 x4 = 3432 5c} 
nours and onehand one [ ~ 52 x 51 x 50 x 49 20825’ : 
The number of deals essentially different is . 
BI: Blint BO 0 ii ini, iat8 wi Bim] £1 
(18 x12 x 11... Sx 2 x1ly 24 
This number, of which the logarithm is 27.34935, is so great, that if 
1,000,000,000 persons, about the population of the earth, were to deal 
the cards incessantly day and night for 100,000,000 years, at the rate of 
a deal by each person a minute, and even if each of these deals were 
essentially different, they would not have exhausted a t5g5oth part of 
the number of essentially different ways in which 52 cards can be distri= 
buted equally between four players. 
12. Let a bag contain 10 slips of paper, each having the name of a 
different individual written upon it, and suppose that the individual whose 
name is drawn is to receive 107. 
In estimating the value of the expectation of these individuals it is 
evident that the sum of the values of all their expectations is equal to 10, 
the sum which one of them must receive, for if they each sold their chance 
of winning the 10. to another person, this person would be sure of receiv- 
ing 10Z.; and since their expectations are equal, for their chances of win- 
ning are the same, if e be the expectation of one of them 
9e= 10, e==1 
that is, the expectation of each is worth 17. Tt is evident, also, that if 
there were m + n slips of paper, each having the name of a different indi- 
vidual written upon it, and if the individual whose name was drawn should Ty 
receive ¢ pounds, the value of the expectation of each individual would be e 
a ; : rr ; 
Era The value of the expectation of two individuals is the sum of the 
m+ n