. ON PROBABILITY. 
w ; . . 
When m+n = 5 w being a whole number ; since is the next whole num- 
ber greater than (m+n) p—q, and since (m+n) p= w, and q is necessarily 
a proper fraction, (m ~~ n) p or w is the next whole number greater than 
(m + n) p — gq, and, therefore, m =w=(m +n) p, n= (m + n) (1 — p) 
=m+n)qg —= that is to say, the event most likely to happen is 
n 
a combination in which the number of repetitions of p and ¢ is proportional 
to the simple probability of the happening of each. 
29. It is only as compared with any other single combination, that the 
one we have just mentioned increases in probability with the number of 
trials : if we estimate the abstract probability of the event corresponding with 
this maximum term, we shali easily find that it diminishes as the number of 
trials increases. - 
For this purpose let the maximum term in m + n trials be represented 
by p.., we have already seen that 
Amt mlnt+n-1),...2.1 _ 
EAB. oa. rT 
where m>e=@m+n)p—q thatis>=m+n+1)p~-1, 
and <(m+n+1)p. 
In one more trial 
apt) (mt n...2.1T , 
dS GY I OT PLL 
m' being limited in the same manner, 
m>=m+n+2)p—1thatis >m+p—-1, 
< (m+n 2)p, thatis <m 4+ p+ 1, 
»~. m/ is either m or m +4 1. 
w mLndl n+ld+m—-—m+n+1)p' 
If m'=m, Crt tot a Pit mtn) en, 
Pm nd n+ 1 
Pw mn 1 i; 
fml= L, mm —— pg =< 1, 
If m z= a - 1 p= < 
.>. in both cases, which will go on occurring successively, py = < pn. k 
30. There is, however, another probability connected with the most proba- “¥ 
ble combination, which does increase continually with the number of trials, for 2 
it is always possible to assign a number of trials, such as to give any required i 
probability that the difference between the ratio of the number of repetitions . 
of the events, and the simple probabilities of the events, shall lie within any 
given limits. Thus, if there are 3 white balls in a bag, and 2 black balls, 
we can always assign a number of trials such as to give any probability as 
near as we please to certainty, that the difference between £ and the ratio of 
the number of white balls drawn to the number of black balls, shall lie be- . 
tween given limits, however near those limits may be assumed. This is a 
theorem of the highest importance in the theory of probability, and indeed, fin 
it is on the converse of it that the value of experience depends. We shall 
endeavour to prove it as shortly as possible, and it will facilitate this object 
if, instead of representing the simple probabilities by p and g, as we have 
: a Dis 4 
hitherto done, we express them by the two fractions —— and ——, in 
a+b a+b 
which @ and 5 are both whole numbers. 7 
i 4 
9)