Page 38 - Applied Statistics And Probability For Engineers

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2-1 SAMPLE SPACES AND EVENTS 21

Message 1
on time late

Message 2

on time late on time late
Message 3

Figure 2-5 Tree
on time late on time late on time late on time late
diagram for three
messages.

One must be cautious with this description of the sample space because there are many more
pairs of items in which both are good than pairs in which both are defective. These differences
must be accounted for when probabilities are computed later in this chapter. Still, this sum-
mary of the sample space will be convenient when conditional probabilities are used later in
this chapter. Also, if there were only one defective part in the batch, there would be fewer
possible outcomes

S 5gg, gd, dg6

because dd would be impossible. For sampling questions, sometimes the most important part
of the solution is an appropriate description of the sample space.
Sample spaces can also be described graphically with tree diagrams. When a sample
ways of
space can be constructed in several steps or stages, we can represent each of the n 1
completing the first step as a branch of a tree. Each of the ways of completing the second step
can be represented as n branches starting from the ends of the original branches, and so forth.
2
EXAMPLE 2-3 Each message in a digital communication system is classified as to whether it is received
within the time specified by the system design. If three messages are classified, use a tree
diagram to represent the sample space of possible outcomes.
Each message can either be received on time or late. The possible results for three mes-
sages can be displayed by eight branches in the tree diagram shown in Fig. 2-5.
EXAMPLE 2-4 An automobile manufacturer provides vehicles equipped with selected options. Each vehicle
is ordered
With or without an automatic transmis- With one of three choices of a stereo
sion system
With or without air-conditioning With one of four exterior colors

If the sample space consists of the set of all possible vehicle types, what is the number of
outcomes in the sample space? The sample space contains 48 outcomes. The tree diagram for
the different types of vehicles is displayed in Fig. 2-6.
EXAMPLE 2-5 Consider an extension of the automobile manufacturer illustration in the previous example in
which another vehicle option is the interior color. There are four choices of interior color: red,
black, blue, or brown. However,
With a red exterior, only a black or red interior can be chosen.
With a white exterior, any interior color can be chosen.