LibraryPirate

3.4 CALCULATING THE PROBABILITY OF AN EVENT 75
Solution:
Using the rule given in Equation 3.4.1, we have
P1E ¨ B2 = P1B2P1E ƒ B2
but, since we have shown that events E and B are independent we may
replace P1E ƒ B2 by P1E2 to obtain, by Equation 3.4.4,
P1E ¨ B2 = P1B2P1E2
= a 40
100 ba40 100 b
= .16
■
Complementary Events Earlier, using the data in Table 3.4.1, we computed the
probability that a person picked at random from the 318 subjects will be an Early age of onset
subject as P1E2 = 141>318 = .4434. We found the probability of a Later age at onset to
be P1L2 = 177>318 = .5566. The sum of these two probabilities we found to be equal to
1. This is true because the events being Early age at onset and being Later age at onset are
complementary events. In general, we may make the following statement about complementary
events. The probability of an event A is equal to 1 minus the probability of its complement,
which is written A, and
(3.4.5)
This follows from the third property of probability since the event, A, and its complement,
A, are mutually exclusive.
EXAMPLE 3.4.8
Suppose that of 1200 admissions to a general hospital during a certain period of time,
750 are private admissions. If we designate these as set A, then A is equal to 1200 minus
750, or 450. We may compute
and
and see that
P1A2 = 1 - P1A2
P1A2 = 750>1200 = .625
P1A2 = 450>1200 = .375
P1A2 = 1 - P1A2
.375 = 1 - .625
.375 = .375
Marginal Probability Earlier we used the term marginal probability to refer to
a probability in which the numerator of the probability is a marginal total from a table
such as Table 3.4.1. For example, when we compute the probability that a person picked
at random from the 318 persons represented in Table 3.4.1 is an Early age of onset
subject, the numerator of the probability is the total number of Early subjects, 141. Thus,
P1E2 = 141>318 = .4434. We may define marginal probability more generally as
follows:
■