Encyclopaedia Britannica > Volume 19, PHY-PRO

probability
774
for all we know of the contents of such an urn is that they are
equally likely to be those of any one of the w + 1 urns above.
If now a great number N of trials of r drawings be made from
such urns, the number of cases where all are white is £+
drawings are made, the number of cases where all are white is
«r+1N; that is, out of the cases where the first r drawings are
white there arejar+1lSr where the (r + l)th is also white ; so that th
probability sought for in the question is
_pr+, _ 1 i-+i + 2’-+1 + 3-+1+ ■ ■ ■
‘a~ Vr ~n lr + 2r + 3r+ . . . nr
16 Let us consider the same question when the hall is not replaced.
First suppose the n balls arranged in a row from A to L as below,
the white on the left, the black on the right, the arrow marking
the point of separation, which point is unknown (as it would be to
a blind man), and is equally likely to be in any of its » + l possible
positions.
A 1 2 B
ooooooooooo.
$
Now if two balls, 1 and 2, are selected at random, the chance
that both are white is the chance of the arrow falling in the dm-
sion 2B of the row. But this chance is the same as that or a third
ball 3 (different from 1 and 2), chosen at random, falling in 2B,—
which chance is because it is equally probable that 1, 2, or o
shall be the last in order. It is easy to see that these chances are
the same if we reflect that, the ball 3 being equally likely to fall m
Al, 12, or 2B, the number of possible positions for the^ arrow m
each division always exceeds by 1 the number of positions for
3 ; therefore as 3 is equally likely to fall in any of the three divi¬
sions, so is the arrow.
The chance that two balls drawn at random shall both be white
is thus J ; in the same way that for three balls is. and so on.
Hence the chance that r balls drawn shall all be white is
;
the same chance for r +1 balls is
1
^rH-r + 2;
thus, as in a large number N of trials the number of cases where the
first r drawn are white is prN, and the number where the first r + 1
are white is jv^N, we have the result:—
If r balls are drawn and all prove to be white, the chance that
the next drawn shall also be white is
_j?r+i_ r+1
^ pr r + 2 '
This result is thus independent of n, the whole number of balls.
This result applies to repeated trials as to any event, provided we
have really no a priori knowledge as to the chance of success or
failure on one trial, so that all values for this chance are equally
likely before the trial or trials. Thus, if we see a stranger hit a
mark four times running, the chance he does so again is f ; or, if a
person, knowing nothing of the water where he is fishing, draws up
a fish each time in four casts of his line, the same is the chance of
his succeeding a fifth time.1
In cases where we know, or rather think we know, the facility
as to a single trial, if the result of a number of trials gives a large
difference in the proportion of successes to failures from what we
should anticipate, this will afford an appreciable presumption that
our assumption as to the facility'was erroneous, as indeed common
sense indicates. If a coin turns up head twenty times running, we
should say the two faces are probably not alike, or that it was not
thrown fairly. We shall see later on, when we come to treat of
the combination of separate probabilities as to the same event, the
method of dealing with such cases (see art. 39).
We will give another example which may be easily solved by
means of (12), or by the simpler process below.
There are n horses in a race, about which I have no knowledge
except that one of the horses A is black ; as to the result of the
race I have only the information that a black horse has certainly
won: to find the chance that this was A—supposing the propor¬
tion of black among racehorses in general to lie p>; i.e., the pro¬
bability that any given horse is black is p.
Suppose a large number N of trials made as to such a case. A
wins in —N of these. Another horse B wins in —N; opt of these
n • " ■ ■ n
1 It may be asked why the above reasoning does not apply to the case of
the chance, of a com which has turned up head r times doing so once more.
The reason is that the antecedent probabilities of the different hypotheses are
not equal. Thus, let a shilling have turned up head once; to find the chance
of its doing so a second time. In formula (12) three hypotheses may be made as
to a double ;—throw 1° two heads, 2° a head and tail, 3° two tails; but the proba-
biuties of these are respectively £, J; therefore by (12) the probability of the
1st after the event is l-Hl+I • l)=i ; that of the second is also 1; and by (13)
the probability of succeeding on a second trial is
, .. , # —
Decause, if hypothesis 2° is the true one, the~second trial must fail.
B is black in _Np. Likewise for C; and so on. Hence the actual
n
case which has occurred is one out of the number
1 1AT
—N q Nn;
n n
and, as of these the cases in which A wins are -N, the required
’ n
chance that A has won is
1
l + (n-l)p'
17. We now proceed to consider the important theorem of Bayes
(see Todhunter, p. 294 ; Laplace, Theorie Analytique des Prob.,
chap. 6), the object of which is to deduce from the experience of a
given number of trials, as to an event which must happen or fail
on each trial, the information thus afforded as to the real facility
of the event in any one trial, which facility is identical with the
proportion of successes out of an infinite number of trials, were it
possible to make them.
Thus we find in the Carlisle Table of Mortality that of 5642
persons aged thirty 1245 died before reaching fifty ; it becomes
then a question how far we can rely on the real facility of the
event, that is, the proportion of mankind aged thirty who die
before fifty not differing from the ratio fftf by more than given
limits of excess or defect. Again, it may be asked, if 5642 (or any
other number of) fresh trials be made, what is the probability that
the number of deaths shall not differ from 1245 by more than a
given deviation ?
The question is equivalent to the following :—An urn contains a
very great number of black and white balls, the proportion of each
being unknown ; if, on drawing m + n balls, m are found white and
n black, to find the probability that the proportion of the numbers
in the urn of each colour lies between given limits.
The question will not be altered if we suppose all the balls
ranged in a line AB (fig. 2), the white ones on the left, the black
on the right, the point
X where they meet
being unknown and
all positions for it in
AB being a priori
equally probable. Then,
m + n points having ^
been chosen at random .
in AB, to are found to * ig. .
fall on AX, n on XB. That is, all we know of X is that it is the
(TO + l)th in order beginning from A of m + n + \ points chosen at
random in AB. If we put AB = 1, AX=x, the number of cases
when the point X falls on the element dx, is measured by
\m + n K ,
. a;w(l - x)ndx,
\m\n
since for a specified set of to points, out of the m + n, falling on
AX, the measure would be xm(l - x)ndx, and the number of such
sets is
\m + n
\m\n'
Now the whole number of cases is given by integrat¬
ing this differential from 1 to 0 ; and the number in which X falls
between given distances a, /3 from A is found by integrating from
/3 to a. Hence the probability that the ratio of the white balls in the
urn to the whole number lies between any two given limits a, 0 is
/:■
xm(l - x)ndx
/:■
(14).
£Cm(l - x)ndx
The curve of frequency for the point X after the event—that is, the
ordinate of which at any point of AB is proportional to the. fre¬
quency or density of the positions of X hr the immediate vicinity
of that point—is
y=x™[l-x)n •,
the maximum ordinate KY occurs at a point K, dividing AB in the
ratio to : n,—the ratio of the total numbers of white and black balls
being thus more likely to be that of the numbers of each actually
drawn than any other.
Let us suppose, for instance, that three white and two black have
been drawn ; to find the chance that the proportion of white balls
is between £ and of the whole ; that is, that it differs by less
than ± £ from f, its most natural value.
a:3(l - xfdx
f x%(l - xfdx
Jo
2256
55
IS ,
= 25 nearly.
18. An event has happened m times and failed n times in m + n
trials. To find the probability that in p + q further trials it shall
happen times and fail q times,—that is, that, p + q more points