ON PROBABILITY.
1 (Ra
—_ — Ri 2 2
(3) +s irtess 2a}
ry 3% de (CHV JIE IN, GED. 2
2)" 2i+1 2.3 =): 3
: 5 y
which probability is greater than 6) , the result when the shilling is sup-
posed homogeneous.
Ex. 20. It follows, from what has preceded, that if an individual has
made m + n assertions, of which m are true and 2 are false ; the probabi-
: : : 1
lity of his telling the truth, in any case, is Gh
) # y Rr far as we draw
our conclusions from these assertions alone,
mol” Wile Ley ov:
Let rats =v, and let p be the a priori probability of an event
which he asserts to have taken place,
The event observed is the assertion by this individual that the event took
place, of which the a priori probability is p.
If the event did take place, the individual tells the truth; the probability
of the event on this hypothesis is pv.
- The probability of the event on the contrary hypothesis is (1 — p) (1=v),
therefore the probability of this hypothesis is
pv
p+ (1 ~p) A —2)
po 1
If tt, |
- pot A-pa-o~ 73
Thus we see that when a witness asserts that an event has taken place,
he renders the probability that it did take place greater than the simple
probability of the event only when his veracity is greater than 4, which result
might have been foreseen. tru
~ Ex. 21. A witness asserts that out of a bag containing a thousand
tickets, a given ticket, say No. 70, has been drawn, required the probability bu
that this number was drawn.
. Let the veracity of the witness be v, as before.
The event observed is the assertion by the witness that the given number
was drawn, the probability of this event, on the hypothesis that the witness
tells the truth, is TE the probability of the event on the contrary hypothe-
wor Lr. 2999 i
sis, namely, that this ticket wasnot drawn, is 1000 (1 —v); but if the witness
is supposed to have no reason or inducement for choosing the No. 70 in
preference to any other of the 999 undrawn numbers, this probability must
1 : , : s :
be multiplied by 999’ which is the probability of the witness choosing this
number from the 999 undrawn numbers, so that the probability of the event
: aa = pon gm fy
on this hypothesis is 005"
3.0.