[ 324 ] 
This is eafily proved in the fame manner with Art. * 
12, 23, 24.. 
That it may appear how far what has been now 
demon ftrated improves the lolution of the prefent 
problem, let us take the fifth cafe mentioned in the 
Appendix to theEfiay, and enquire what reafon there 
is for judging that the probability of an event concern- 
ing which nothing is known, but that it has happen- 
ed 100 times and failed 1000 times in 1100 trials, 
lies between ^ -f- — and — — . The fecond 
H 1 I 10 II IIO 
rule as given in Art. 12. informs us, that the chance* 
for this mud lie between .6512, (or the odds of 186 
to 100) and .7700, (or the odds of 334 to 100). But 
from the laft Art. it will appear that the required 
chance in this cafe muft lie between 2 E, and 
P t ~> P 1% 
I — E a b — 2 E a b 
2 + 2 X 
i -j- 4 o- IL a $ -j- E a l? 
IO n 
or. 
between 
.6748 and .70 57 ; that is, between the odds of 239 
to 100, and 207 to 100. 
In all cafes when 2; is fmall, and alfo whenever 
the difparity between p and q is not great 2 £ is almoft 
exactly the true chance required. And I have reafon 
to think, that even in all other cafes, 2 £ gives the 
* In the Appendix, this chance, as difcovered by Mr. Eayes’s 
fecond rule, is given wrong, in confequence of making m 1 equal 
n 3 _ „ 3 
to — , whereas it fhould have been taken equal to 
P 9 ip q 
as appears from Article 8. 
true 
