Page 44 - Applied statistics and probability for engineers

22 Chapter 2/Probability

Hospital

1 2 3 4 Total
Total 5292 6991 5640 4329 22,252

LWBS 195 270 246 242 953

Admitted 1277 1558 666 984 4485
Not admitted 3820 5163 4728 3103 16,814

Diagrams are often used to portray relationships between sets, and these diagrams are also
used to describe relationships between events. We can use Venn diagrams to represent a sample
space and events in a sample space. For example, in Fig. 2-8(a) the sample space of the random
experiment is represented as the points in the rectangle S. The events A and B are the subsets of
points in the indicated regions. Figs. 2-8(b) to 2-8(d) illustrate additional joint events. Fig. 2-9
illustrates two events with no common outcomes.
Mutually
Exclusive Two events, denoted as E 1 and E 2 , such that
Events
E 1 ∩ E 2 = ∅
are said to be mutually exclusive.

Additional results involving events are summarized in the following. The deinition of the

complement of an event implies that
E
( ′ ′ = E
)
The distributive law for set operations implies that
( A∪ B)∩ C = ( A∩ C)∪( B ∩ C) and ( A∩ B)∪ C = ( A∪ C)∩( B ∪ C)
DeMorgan’s laws imply that
( A∪ B)′ = A′ ∩ B′ and ( A∩ B)′ = A′ ∪ B′

Also, remember that
A∩ B = B ∩ A and A∪ B = B ∪ A

2-1.4 COUNTING TECHNIQUES

In many of the examples in this chapter, it is easy to determine the number of outcomes in each
event. In more complicated examples, determining the outcomes in the sample space (or an
event) becomes more dificult. Instead, counts of the numbers of outcomes in the sample space

and various events are used to analyze the random experiments. These methods are referred to
as counting techniques. Some simple rules can be used to simplify the calculations.
In Example 2-4, an automobile manufacturer provides vehicles equipped with selected
options. Each vehicle is ordered
r With or without an automatic transmission
r With or without a sunroof
r With one of three choices of a stereo system
r With one of four exterior colors