Page 41 - Applied statistics and probability for engineers

Section 2-1/Sample Spaces and Events 19

S = { yy, yn,ny,nn}

If we are interested only in the number of conforming cameras in the sample, we might summarize the sample space as
S = {0 1
, , } 2

As another example, consider an experiment in which cameras are tested unitl the lash recycle time fails to meet

the speciications. The sample space can be represented as
S = { n, yn, yyn, yyyn, yyyyn,and so forth }
and this is an example of a discrete sample space that is countably ini nite.

Sample spaces can also be described graphically with tree diagrams. When a sample space can
be constructed in several steps or stages, we can represent each of the n 1 ways of completing the

irst step as a branch of a tree. Each of the ways of completing the second step can be represented
as n 2 branches starting from the ends of the original branches, and so forth.

Example 2-3 Message Delays Each message in a digital communication system is classiied as to whether it

is received within the time speciied by the system design. If three messages are classiied, use a

tree diagram to represent the sample space of possible outcomes.
Each message can be received either on time or late. The possible results for three messages can be displayed by
eight branches in the tree diagram shown in Fig. 2-5.
Practical Interpretation: A tree diagram can effectively represent a sample space. Even if a tree becomes too large to
construct, it can still conceptually clarify the sample space.
Message 1

On time Late
Message 2

On time Late On time Late

Message 3

On time Late On time Late On time Late On time Late

FIGURE 2-5 Tree diagram for three messages.

Example 2-4 Automobile Options 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
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.