26 ON PROBABILITY. 
It is worth observing that these conclusions would not be affected by retain= 
ing all the four hypotheses which might have been made before the observa- 
tion. For the probability of the observed event on the hypothesis we have 
rejected, namely, that all three balls are white, being = 0, the probability 
of the others will not be altered by including it also in the sum of probabili- 
ties which make up the denominator of the above-written fractions." 
47. Let the ratio of the white balls to the whole number of balls be any of 
the following quantities,’ 
let any of these hypotheses be equally probable d priori, and let a white ball 
have been drawn, 
The probability of the event observed, namely, the drawing a white ball, 
on the first hypothesis is , the & priori probability of this hypothesis is - 
| 
because there are i hypotheses equally probable ; therefore the probability of 
this hypothesis is 
ry 
é 2 
1 ETT 
se i1+248...4} % (rt 1) 
Tht ve ‘rsd po 0 : 
Similarly, the probability of the second hypothesis is GD of the third 
A 5%2 and so on 
i(Gi-++1)° 
The probability of drawing a white ball in a future trial, after replacing 
the ball drawn, if the first hypothesis were the true one, would be z; the 
probability of this hypothesis is C+D} therefore, the probability of 
drawing a white ball, considering only this hypothesis, is TT 
. ts Te : ial Da .28 
Similarly, considering the second hypothesis, the probability is TCL 
2 
2 
third Sax. 
ii41 
and the sum of all these, or the probability of drawing a white ball again, is 
22 
errr ll... 401, 
orn iltEt + 21 
In order to find the sum of this series, we shall employ a method of 
greater generality than is necessary in this particular case, because it fur- 
nishes the readiest method of summing a great many series of the same 
kind. 
a? 
7 — + &c. 
¢ Latin Cc 
2 yo 22 : 
fom 1+ 224+ 2XE 4 ge 
an _-