More
general
application.
The series
that are
given in
Anthropometry
are
analogous
to simple
events.
Introduction
of a few
necessary
algebraic
symbols.
q are respectively the chances that the event will, and will not, occur. As the event must
do one or the other p + q = i denotes certainty. The odds in favour of the event are m to
n — m, those against are n — m to m. In the example already given m= i, n — m — 51,
« = 52: so / = j j i s the chance of drawing the ace of spades, and y = f^ is the chance of
not drawing it. The odds against this event are 51 to 1 *.
In Anthropometrical Series the jf events ’ are clearly independent, for one skull cannot
influence another so as to affect its measurements. We may therefore confine our
attention to independent events. The series we consider are simple combinations of
independent events, and are exactly analogous to such series as may be obtained by
repeating any operation a number of times.
The operation which we take for comparison is that of placing a number of pennies at
random on a table. The a priori chance of any particular penny turning up ‘ heads ’ is the
same as its a priori chance of turning up ‘ tails.’ In the same way in such series as
those with which we are now dealing, errors in observation, or deviations from a type, are
generally assumed, in the absence of any specially noted causes of disturbance, to tend
equally to excess and deficiency (on this point see Chap. VIII, opening paragraphs). This,
as will be shown later, much simplifies the formulae which we use and the methods by which
we arrive at them.
There are a few algebraic symbols necessary in the study of the theory which it
will be convenient to introduce here. Suppose we have a number, n, of pennies. Then if
we take t of them, where t is any number not greater than n, we denote by "Ct the number
of ways in which it is possible to pick up t pennies out of the n : e. g. if there are three
pennies (that is to say n = 3) we can pick up two of them in three ways : that is to say, if
t= 2, ”Ct is 3. The number of ways in which the three pennies can be placed on the
table is eight, as any reader can verify for himself. Each of these eight occurrences is of
equal probability, and consequently the chance that, on placing the pennies on the table,
two of them shall be heads uppermost is f.
The principle can be extended to any number of pennies; the fraction which
denotes the chance of t out of n pennies being heads uppermost will have for its numerator
the expression MCt and for its denominator 2“. In the case which we have quoted for an
example nCt is 3 and 2” is 8 f.
* T h e examples here given are illustrative o f simple .events which are independent o f one another. When,
however, an event is dependent upon the occurrence or non-occurrence o f some other event the rule is naturally
more complex. I t may be epitomized in the following fo rm :— I f p 1 is the chance o f one event and p % is the chance
o f a second event subsequent to the first, then the chance that the two w ill happen successively and in the prescribed
order is p 1 x / 2.
F o r instance, what is the probability that the ace o f spades will be drawn from a pack o f cards and that
the next card to b e drawn will be the ac e o f hearts? From the application o f the rule it will be evident that
as the chance o f drawing the ace o f spades to begin with was fa , and the chance o f drawing the ace o f hearts from
the p ack diminished b y one card was fa , the chance o f the double event is f a x fa .
Such complex occurrences, however, are not amongst the problems which are dealt with in our present
investigation.
+ T h e case in which the chances are unequal a p r io r i may b e treated in the same manner, but the reasoning
is more complicated. ■
A convenient example o f this is to b e found in dice-throwing. He re the chance o f throwing a six with
one die is | and the chance against § . I t can b e shown b y the same methods as those hinted at above that the
nC
chance o f throwing t s ixes with n dice is -^ j 5"“ *.
In cases analogous to the above where the number of pennies is small it is a com- More
paratively simple matter to calculate the probabilities of a given occurrence bv arithmetic, general
, , e . . . . 1 * 1 , . . . . * application.
Dut ior the long series which commonly occur in statistics a less clumsy method is needful,
and one moreover which can be applied to series even when the numerical values of n and
t are unknown. Such a method is supplied with the help of the Binomial Theorem.
This theorem deals with the expansion of a certain algebraical expression in a series The
of terms, and the law according to which these terms are calculated is intimately connected Theorem in
with the Law of Probability. its relation to
The part of this theorem with which we are directly concerned is that in which Probabil,ty*
it deals with the expansion in a series of terms of an expression such as a + x, where that
expression is multiplied by itself a certain number of times, say n.
For the moment we may still further simplify matters by using the number one in the
place of a. We have, then, under consideration the expression which results from the
expansion into a series of (i +•#)”. The resulting series is one composed of a number of
terms, each consisting of a power of x multiplied by a constant which varies with the
power of x considered. If, for example, n = 3 we have (1 +x)3= 1 + $ x+ 3 X 2 -hx3. The
constant connected with x *, or the i th power of x, is what we have already denoted by nCt.
Consequently the chance of any particular number of heads occurring in our experiment
is the same as the coefficient of the corresponding power of x in the expansion of
- i » e-g- the chance of two of three pennies showing heads is the coefficient of x 2 in
- ~^~3 \ and so is §• as already shown.
In the expansion of ( 1 +x)n the coefficients form a gradual progression rising from the Properties
lowest values at the beginning until the middle term, if n is even, or the two middle terms, coefficients
if n is odd, where the coefficients have their maximum value. From this point the in ^
coefficients begin to decrease steadily, and the terms at equal distances on either side of +?)".
the middle term or terms have equal coefficients. This may be seen in any particular
example as, for instance, where n = 3, as shown above, or where n = 6 when
(1 +x)6= i+ 6 x + i$ x l+20Xi+ i$x4+ 6 xB+ x6.
In the general case nCt, i. e. the coefficient of x* in the expansion of (1 + x)n, may be
expressed as a fraction, the numerator of which consists of the product of t factors differing
from one another by unity and ranging from n - t+ 1 to n, and the denominator consists
of the product of the first t numbers. Thus ”Ct stands for
n (n - 1) ( n - 2) - • • (n -1+ 1)
1 . 2 . 3 . ••• t
and (1 +x)n = 1 + n x ^ r ^ — —x*+ I S fs ffi
1 . 2 .
....... i . 2
To show this more clearly we may take Quetelet’s example in which « = 999. The
first coefficient is 1, the second 999» the third j x and so on to the highest point,
which is reached in the 500th and 501st terms, which are equal, being respectively
999 • 998.997 • ‘ • (999 ~ 499 + 1)
1 . 2 . 3 . ----- 499
and 999- 998-997 -••■ (999- 498) (999-50 0 + 1)
i • 2 . 3 • 499 • 500
MACIVBR K