; 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.