32 ON PROBABILITY. 
hence we see how prodigiously the probability of an event of this kind is 
increased by the concurrent testimony of many individuals. It must, how- 
ever, be remarked, that the weight of the concurrent testimony of different 
individuals depends entirely upon the absence of inducement to lead these 
individuals to choose any one particular number, and upon the absence of 
collusion. 
50. If we have no data to determine the veracity of an individual, and if one 
he asserts an -event to have taken place, of which the simple probability is ) 
P» in order to find the probability that the event took place, we must consi- a” 
der the probability of the event upon every hypothesis which can be formed; e 
that is, we must suppose all values of his veracity between zero and unity 
to be equally probable, & priori. The probability, therefore, that the event the 
did take place ie: 
=/ pvdo =f; prvdo hes 
J pr+ A —0)(1-p) 1—p+4@p- D7 oh 
taken between the limits vy = 0 and v = 1; foe 
mp {Rp—1Nov+1—p—(1—-p) ldo ilo 
T2p—1 1-p4+Q@p-Dv to 
con: 
Pp : 1-7 i 
Es TV gg. (0—p + @Rp—1)v ! c vies 
291 2p ~ 1° rar I+ cien 
est Zp 1 —— Ley loo ll 3 
2p —11| 29-1 "=\0—-p/f 
When p = , this is equal t 2h log 9 | = -s16363 bout — 
en p = 5» this is equa oz glog 9 = of about ou, 
Ex. 24. A jury consists of 7 individuals ; let the probability of each sepa= 
rately giving a right decision be p, what is the probability that a unanimous 
decision is a correct one? Two hypotheses can be formed, namely, that the 
decision is a correct one, or the contrary; the event observed is a unani- 
mous decision, and the @ priori probability of this event on the first hypo- 
thesis is p”, the d priori probability of the event on the second hypothesis is and J 
¢ ee iis 2" (T+ 
(1 — p)*, therefore the ‘probability of the first hypothesis is FEU—p)" evel 
1 1 wie 
which is greater than 3 only when p > 3 Therefore it is probable that a 
unanimous verdict is a correct one, only when it is probable that each jury- i 
man considered separately will give a correct decision, The same rule holds Be 
a fortiori, when the verdict has been given by a majority only. Jt 
Tees 9 ou 
If p= -9 and n = 12, this probability is equal to IT dri 
A jury composed of n — 2 m individuals is correct. Xz 
Similarly, the probability that the decision of a majority is correct allt 
_ptap tA =p +n. (n-1) 2 ~pP 4... li 
re 7% n MN ~~ N ~— al le 
F+d—-py+tup.(l 5. {py +0 —2r"] + &, tir 
The probability that a decision given by n — 1 is correct, is similarly - 
Sad LD) me ms — Te Ott 
Fld -ta(l=9)""'" FL (1 ~py} 
and generally the probability that a decision giverr by # — m of the jury is 
correct, is the same as the probability that a unanimous decision of