ON PROBABILITY. 33 
a a jury composed of 7 — 2m individuals is a correct one. If n= 12 and 
p = "9. the probability that a decision of a majority is a correct one by 
| . 999458768178 _ 19519 
ho the preceding expression = 999508948516 — 19530 nearly. 
If pis unknown, the probability that a unanimous decision is a correct 
dual, aq i¢ : pe 
a one must be found by taking the integral f a TEP between the 
vi 7 +l ~7) 
Wi limits through which p may be supposed to vary, multiplied by J dp taken 
od ® between the same limits. 
a hy 51. The decision of the jury in this country can only be considered as 
+(e event that of a simple majority, and the probability that this decision is a correct 
one is small, unless the simple probability that each juryman separately 
gives a correct one, is taken to be very great. If this probability is 2, the 
probability of a correct decision is very little greater than 12. The simple 
probability of any juryman giving a correct decision cannot be supposed to 
be strictly the same for each juryman composing the same jury, and it must 
also depend very much upon the natare of the question which is submitted 
to his determination. As this probability rests ouly on conjecture, we have 
considered the preceding questions relating to the decision of a jury with a 
view of showing how they might be solved if we were in possession of suffi- 
cient data rather than as laying any stress on the results obtained. 
52. A bag contains a number of balls of ¢ different colours: 
m, of the Ist colour have been drawn and replaced, 
} My; 2nd, 
bout —, Mg 
L inom +m; ...... -}m; trials; required the probability of drawing 
‘each coma mn, balls of the first colour, n, of the second, n; of the 4 colour, in 
or nm ~ng.... + mn; succeeding trials. 
Let x, be the a priori probability of drawing a ball of the 1st colour, 
2, 2 » » 2nd, 
| &; tH) ” i) wh, 
and let C be the coefficient of x; ™ x 2, ....z,™ in the developement of 
(Ti+ 2....Fx)™+™--:F™; then the probability of the observed 
event is CO Xa. wm, "en" the probability of the hypothesis that 
| #, is the probability of drawing a ball of the 1st colour, 
hie fr Lg 3» a 2nd, 
ea x; 2 25 i, 
each Jury isCXx™ Xx" X...z;™ divided by the sum of all the values of which 
> rule holds this fraction is susceptible ; and if C, is the coefficient of 7, ™ x DTN, 
in the developement of (#, + @y...-a) +" +m the probability of 
drawing n, balls of the 1st colour, n, balls of the 2nd colour, n; balls of the 
i" colour is the sum of all the values of which the quantity C x C, 
Xa Mtmip, Mtn) | lg ™ +5) js susceptible, divided by the sum of 
» all the values of which the quantity C x #,™ X x," .... x; ™ is susceptible, 
Re Itz, @,.... x;, be supposed to vary from # = 0 to x = 1, and ail these 
ak values are equally probable a priori, then the probability required is found by 
+ fe. taking the integral 
amilarly I Tp xg, dd LL tat dr dad 2; 
between thelimitsz, =0, 0 =l—-0~2,........ = 2 
. i 2.=0, Bi, =l-0,—-g—~a2,.... — 2, 
i) %..=U La=1l—a—-5........ —_ 2.3 
ie & =0 “pl =] 
D