ON PROBABILITY: 
a our probability 
_ d.(sum of other parcels) + 4- «,, i 
i (i + 1) (sum of other parcels) + k + a,, > I 
since every proper fraction is diminished by taking the same quantity (in this 
instance % + a,,) from its numerator and denominator. 
37. We thus have a probability, > ITV that the number of recurrences 
: of the event, whose probability is = will, in m, (@ + b) trials, lie between 
the numbers (m a + m), (m a—m), however great ¢ be taken, and the limits 
rm} of the ratio of the number of repetitions of this event to the number of trials 
tl.mab ma + m ma —m a+1 a—1 
———— d ee ts ee 9 ° ° 
i» are ESS and —~ @Tb or = TI and ZIT the difference of whichis 
40F older c 
re 5 and since the only restriction on the value of @ and b is, that they 
must be in the proportion of p to ¢, we may increase a -- b at pleasure, and 
thus bring the limits of this ratio as close as we please, and yet have a proba- 
bility, > fo that the observed ratio will be between them. If all that is 
required is that there will not be fewer than m ¢ — m recurrences, the pro- 
bability clearly becomes much greater, as also the probability that there will 
not be more than m a + m. 
* It only remains to give an example to show fully how this theorem is 
applied. 
Ex. 15. A bag contains three white balls and two black balls, from which a 
ball is repeatedly drawn and replaced. Required the number of trials in 
ima) which the odds will be at least 1000 to 1 that the ratio of the number of 
Lomb times that a white ball is drawn to the whole number of drawings is not less 
29 5 
than 50’ and not greater than 50° 
The value of ¢ is here 1000 ; @ = 30, and b == 20. 
i log. i + log. (@ — 1) _ log. 1000 + log. 29 
TT log. (a+ 1) —log.a log. 31 — log. 30 
4:462398 
ar b I een b ” 
r become Soil S13 <3 
Therefore we must take m = 314, and the number of trials = m (& + b) 
= 15700, in which number of drawings the odds will be more than 1000 to 
1 that a white ball will have been drawn not less than m ¢ — m, or 9106 
times, and not more than m a + m, or 9734 times. 
If the odds had been required to be at least 10,000 to 1, or 100,000 to 1, 
we need only change the value of 7, which gives, in the first case, 
5°462398 
and in the second case, 
6°462398 
EE nd b) = 22700, 
M = oo Tia10d < 454 and m (a + b) = 
23