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