Statistical Methods in Medical Research 4ed

52 Probability
In general, pairs of events need not be independent, and the general form of
the multiplication rule (3.2) must be used. In the earlier example we referred to
the probability of a doctor being male and having qualified in England. If these
events were independent, we could calculate this as
0 8 0 6 ˆ 0 48,
and, denoting the events by A and B, (3.1) would give
P…A or B or both† ˆ0 80 ‡ 0 60 0 48
ˆ 0 92:
These events may not be independent, however, since some medical schools are
more likely to accept women than others. The correct value for P(A and B) could
only be ascertained by direct investigation.
As another example of the lack of independence, suppose that in a certain
large community 30% of individuals have blue eyes. Then
P…blue right eye† ˆ0 3
P…blue left eye† ˆ0 3:
P(blue right eye and blue left eye) is not given by
P…blue right eye† P…blue left eye† ˆ0 09:
It is obtained by the general formula (3.2) as
P…blue right eye† P…blue left eye; given blue right eye†
ˆ 0 3
ˆ 0 3:
1 0
Addition and multiplication may be combined in the same calculation. In the
double sequence with a coin and a dice, what is the probability of getting either
heads and 2 or tails and 4? Each of these combinations has a probability of 1
12 (by
the multiplication rule for independent events). Each combination is a possible
outcome in the double sequence and the outcomes are mutually exclusive. The
two probabilities of 1
12
may therefore be added to give a final probability of 1
6 that
either one or the other combination occurs.
As a slightly more complicated example, consider the sex composition
of families of four children. As an approximation, let us assume that the
proportion of males at birth is 0 51, that all the children in the families may
be considered as independent random selections from a sequence in which the
probability of a boy is 0 51, and that the question relates to all liveborn infants so
that differential survival does not concern us. The question is, what are the
probabilities that a family of four contains no boys, one boy, two boys, three
boys and four boys?