; ON PROBABILITY. 
- is with respect to the colour of No. 2. 
No.l. No.2. No.3. 
1st hypothesis, black, black, white. 
2nd ‘ black, white, white. 
| On the first hypothesis No. 3 must have been drawn; we have, therefore, 
" only one case which favours this hypothesis. On the second hypothesis, 
Rua. either No. 2 or No. 3 has been drawn, so that we have two cases which 
CORSE: To, favour this hypothesis, and, therefore, the probabilities of these hypotheses 
Or of the 1 2 1 2 
respectively are —— and ——, or — and —. 
alli SEAT ET a Ee ee 
i Se In order to extend the principle of this reasoning to the general case, Jet 
rm us suppose that an event has been observed which must have resulted from 
Mek y one of a given number of causes. Let the probability of the existence of 
Py one of these causes have been estimated at P before the observed event took 
king place, and let the probability of the observed event calculated upon the cer- 
wd titude of that cause be called p. 
3% ok The probability that the event will happen in consequence of that cause 
= P p, and the probability that the event will happen, without reference to 
any particular cause = =. P p ; extending the sign of summation to all the 
possible causes. 
The probability that the event will happen in consequence of the selected 
cause, (or Pp), may be estimated in a different manner : it equals the pro- 
duct of the probability that it will happen, (or =. Pp), by the probability 
ee: that if it does happen, it will be in consequence of that cause : the latter is 
ep evidently the probability of the selected cause derived from the observation. 
P 
the whale’ Therefore the derived probability of that cause = iy which result 
1S long may be stated in the following important theorem. 
haa; 1 The probability of any hypothesis is the probability of the observed 
Se os event upon this hypothesis multiplied by the probability of the hypo- 
Me prog thesis antecedently to the observation divided by the sum of the products 
isi which are formed in the same manner from all the hypotheses. : 
hie a0 46. The probability antecedent to the observations under consideration is 
4 0 Pash called the a priori probability ; but in using this term it must be remem- 
rn bered that it is relative to a given epoch. 
arto these Ex. 17. . Thus in the instance of a bag containing three balls, of 
4 De repre: which a white one has been once drawn and replaced. There are three 
a Dags con possible suppositions : first, all three are white ; second, two only are white; 
! resus We third, only one is white. On the first supposition the probability of the event ob- 
fil the ol served is certainty or 1, on the second the probability is %, on the third it is + 
§ In Lucile > : 1 
straight, Therefore the probability of the first hypothesis = — prEp Et, 
tanced is 1p 3 Le 5 
oh den 3 1 
second ,, = a 2 = 1 = 3? 
b FS Ts 
} 1 
3 
toird > BE eer eer - 0 2 
14 2 * ¥Y 6 
33 
25 
1.