Statistical Methods in Medical Research 4ed

n
ˆ 1: …3:11†
0
This is clearly the correct result, since there is precisely one way of selecting 0
objects out of n to be labelled as As: namely to select all the n objects to be
labelled as Bs. Note that (3.11) accords with (3.9) if we agree to call 0! ˆ 1; this is
merely a convention since 0! is strictly not covered by our previous definition of
the factorial, but it provides a useful extension of the definition which is used
generally in mathematics.
The binomial coefficients required in the example of §3.2 could have been
obtained from (3.8) as follows:
4
0
4
1
4
2
4
3
4
4
ˆ 1
4
ˆ ˆ 4
1
4 3
ˆ ˆ 6
1 2
4 3 2
ˆ ˆ 4
1 2 3
4 3 2 1
ˆ ˆ 1:
1 2 3 4
A useful way to obtain binomial coefficients for small values of n, without any
multiplication, is by means of Pascal's triangle:
n 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
etc. etc.
In this triangle of numbers, which can be extended downwards indefinitely, each
entry is obtained as the sum of the two adjacent numbers on the line above.
Thus, in the fifth row (for n ˆ 4),
4 ˆ 1 ‡ 3, 6 ˆ 3 ‡ 3, etc:
3.6The binomial distribution 67
Along each row are the binomial coefficients
n
,
0
n
, ...
1
up to
n
,
n 1
n
n :
The probability that the sample of n individuals contains rAsandn
then, is
rBs,