ON PROBABILITY. 19 
“L The 1 
if p= B= 2 the expectation of loss =2¢2 a + 2¢ (i — 1) 225. The 
expectation of a player, who is entirely ignorant of the value of p, is found 
by integrating the expression here found 
.(1—1 
Siia.@p-1)- 00 0p 1p}, 
3ia.Rp—-12—-i.G~1).b.@2p~1) 
— 12 
and from p= 0top=1, = — 2 =D 2 
’ It should be observed that this solution only applies to the case when 
there is a limiting equation between a and b, such that a—(i—1)b >0, 
otherwise there might be a conjunction in the game, in which the player could 
; not follow the rules of this scheme, and consequently would alter his expec- 
: tation. If this be not attended to, the theorem supposes, what can never 
take place in practice, that the player has the power of reducing his stake be- 
low zero, that is, of taking his adversary’s situation in some point of the game. 
28. Let there be two conflicting events P and Q, of which the probabi- 
1)» lities are p and q respectively, and let m +n trials take place; the proba- 
bility that the event P will happen m times, and the event Q, 7 times, 
§t mr without regard to the order in which they succeed each other, is 
us (rn). neml). v2.1 0 
WE reer eee I, O°, 
1.2 coomab.8 . np 
which we shall represent by p,. Similarly, the probability that the event P 
will happen 72 + 1 times and the event Q, (zn — 1) times, is 
min). mt pwd) yun2.1 0, la 
( ( Pp +1 q 1 = Pm ore 
¥.2..(n4+131.2,. (n+1)* 
Fo m+ 1). : 7 
UW Det => Pips Ds: >1:19> whe, and since p 4 ¢ = 1, 
m > (m + n) p — ¢, and this continues until m = (m + 2) p — ¢, in which 
case p,, = Pp; and, since m must be a whole number, the greatest term 
(1p) in the developement of (p~- ¢)"*" is that when m = (m + 2) p — gq, if 
(m +n) p — q be a whole number, or, if not a whole number, the next 
greater, 
1 putt 2 1 
(1-2 Suppose, for instance, p = 3 and consequently g =, 2 and let m | n 
g of all the - 
1 the ind == 17x 7218 kf of aul or = = 1; so that the most probable event, as 
p R i= ] 
compared with any other event which can occur in 17 trials, is a repetition 
of P 11 times, and Q 6 times. 
. 18 x 2 —1 
~ th If m + n = 18, m is the whole number next greater than EER 
~ 1h go E2 ; which is 12; and the most probable event as compared with any other 
hi which can happen in 18 trials, is a repetition of P 12 times, and of Q 
Aha 6 times, 
cD