ON PROBABILITY. = 
: "ap a. b 
31. The events of which the probabilities are .——— —— would be 
a+b a-4 0b 
repeated in m (ab) trials exactly ma and mb times, if they were com- 
bined in the ratio of their simple probabilities. Let us suppose that the 
number of times that the first will be repeated in the observed event lies 
between the limits ma — m and m a + m, that is to say, that there will not 
be fewer than ma — m and not more than m a + m recurrences of that 
event in the m (a + b) trials. The probability that this will be the case is 
: : - a bh \mlakh 
the sum of the 2m + 1 terms in the developement of (+m) 
from the term whose argument is "=" to the term whose argument is 
@"**™, both inclusive. We shall call these two last-mentioned terms the 
first and second limiting terms, and it is clear that the maximum term, the 
argument of which is ¢™¢, lies between them, 
32. The whole series may thus be written out at length : 
a b m (a -- b) 
GG -+- =) = Pm(atb) te Patatiiml os 05:02 + Prtatom~1 
First limiting term, Maximum, » + Second limiting term, 
+ Donati alo +t Pinas] + Pine i Pma-1 °85 000 1 Dna mm 
Parcel of the first limit. Parcel of the second limit, 
~ Dreema1 + maaead iA i inh + Pos 
the index at the foot of p always denoting the power to which « is involved 
in that term. Our object is to show that the 2m -- 1 terms within the 
limits, may be made as many times greater than the vest of the series as we 
please ; and we shall do this by showing that the first m of these 2m + 1 
terms, which we will call the parcel of the first limit, can be made as many 
times greater as we please than all which precede them, and the last m 
= terms, or the parcel of the second limit, as many times greater as we please 
than all which follow them. 
33. There are mb — m terms which precede our first limit, which may 
be classed in (b — 1) parcels, each containing m successive terms, and simi- 
Je larly the m @ — m terms, after the second limit, may be classed in (¢ — 1) 
st probe parcels, each containing m successive terms. As the maximum term p,, 
Stil, for is in the middle of our limits, and as the values of the terms increase from 
required each end of the series up to the maximum term, the sum of all the b—-1) 
y! ons parcels before the first limit will be less than (b — 1) times the parcel next 
eel 7 before the first limit ; and the sum of all the (a — 1) parcels beyond the second 
Ni 2 limit will be less than (« — 1) times the parcel next following the second 
as limit. It is also plain, for the same reason, that the parcel of the first limit 
abil : is greater than the parcel which immediately precedes the first limit, and, 
Je gl} since the ratio of the maximum term to the first limiting term, or the ratio 
ha 4 Of Pua tO Prim, is less than the ratio of any term in the parcel of the first 
In 5 limit, to the corresponding term in the parcel next preceding the first limit, 
od idee (because these ratios continually approach nearer to a ratio of equality as 
Wie ge they approach the maximum term,) it follows that the ratio of p,., to Deen 
0s oe is less than the ratio of the whole parcel of the first limit to the whole par- 
2 RY cel immediately preceding it. 
34. If, therefore, we can show that p,, can, by a proper assumption of 
m, be made greater than ¢.(b — 1) times Pmat+m however great 7 is taken, 
it will follow that this value of m will make the parcel of the first limit still 
)