ON PROBABILITY. 31 
The probability, therefore, of the first hypothesis is TTT or v, the 
veracity of the witness. 
Ex. 22. Two individuals, whose veracities are » and v/, assert that an 
event has taken place, of which the probability is p. 
Two hypotheses are admissible, namely, that the event did take place, 
and that both the individuals tell the truth; or, that the event did not take 
place, and that both individuals lie; the probabilities of the assertions on 
fivighe yg these hypotheses are vv'p and (1 —v) (1 =) (1 — p), therefore the 
he proba probabilities of these hypotheses are 
i oVp and Lm D0 ~ 2) (1 ~ 2) 
8% du 2p + (A=) (=v) A =p) vx? x p+ (1—v) (1=v) (1p) 
respectively. i 
So if 7 individuals, whose veracities are v,, Vy Vg, V,, assert the event to 
Ren have taken place, the probability that it did take place is 
V1 Vy Vy ® ss 0» WO, P 
7 x ETT TT TT TTT Tae 
| Vive pF (l=) (I=) .....(1 —0,) 1 =p 
eee ol If n + 1 individuals assert the event to have taken place, the probability 
N that it did take place is 
¥ probability 
V1 VgVgs ovens sVpgP 
«p) (L=1), VV... Vppupt+ A =v) A —2).....( — Vg) (1 — p) 
1 
which is greater than the former probability if v,,; > 5 S0 that the assertion 
of the » + 1* individual increases the probability of the event arising from 
the testimony of the other n individuals, only when his veracity is greater 
than 1. 
Ex. 23. Two individuals, whose veracities are » and v/, assert that a given 
aken place, ticket has been drawn out of a bag containing a thousand tickets, The pro- 
1 the simple bability of the event on the hypothesis, that both the individuals tell the 
hich result ) 
o truth and that this ball was drawn, is Too the a priori proba- 
hy bility, that both the individuals lie and that the given ball was not drawn, is 
999 
1—=2) 1 =) 1600." if, however, these individuals have no inducement 
¥ to choose the given ticket amongst the undrawn numbers, this probability 
must be multiplied by TF which is the probability of their both select- 
ng the same number amongst the undrawn numbers; the probability, there- 
fore, of the first hypothesis, namely, that the given ticket was drawn, is 
2 
2+ (l=) (AQ =o) 1° 
999 
If v,, vs, vs .. . », are the veracities of n individuals, who all assert"that 
a given ticket was drawn, the probability that the given ticket was drawn is 
ia VV V3... ) 
D102 Use 0 0p + (1 = 2) (1 — V3) ..e(l—2,) 1 a : 
(999)"=1