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18 CHAPTER 2 PROBABILITY

Call 1 2 3 4

Call duration
Time 0 5 10 15 20
Current Call 1 2 Minutes

3
Call 3 blocked
Call duration
Time 0 5 10 15 20
Voltage Minutes
Figure 2-3 A closer examination of the system Figure 2-4 Variation causes disruptions in the system.
identifies deviations from the model.

uncontrollable inputs, the same settings for the controllable inputs do not result in identical
outputs every time the system is measured.

Definition
An experiment that can result in different outcomes, even though it is repeated in the
same manner every time, is called a random experiment.

For the example of measuring current in a copper wire, our model for the system might
simply be Ohm’s law. Because of uncontrollable inputs, variations in measurements of current
are expected. Ohm’s law might be a suitable approximation. However, if the variations are
large relative to the intended use of the device under study, we might need to extend our model
to include the variation. See Fig. 2-3.
As another example, in the design of a communication system, such as a computer or
voice communication network, the information capacity available to service individuals using
the network is an important design consideration. For voice communication, sufficient
external lines need to be purchased from the phone company to meet the requirements of a
business. Assuming each line can carry only a single conversation, how many lines should be
purchased? If too few lines are purchased, calls can be delayed or lost. The purchase of too
many lines increases costs. Increasingly, design and product development is required to meet
customer requirements at a competitive cost.
In the design of the voice communication system, a model is needed for the number of calls
and the duration of calls. Even knowing that on average, calls occur every five minutes and that
they last five minutes is not sufficient. If calls arrived precisely at five-minute intervals and lasted
for precisely five minutes, one phone line would be sufficient. However, the slightest variation in
call number or duration would result in some calls being blocked by others. See Fig. 2-4. A system
designed without considering variation will be woefully inadequate for practical use. Our model
for the number and duration of calls needs to include variation as an integral component. An
analysis of models including variation is important for the design of the phone system.

2-1.2 Sample Spaces

To model and analyze a random experiment, we must understand the set of possible out-
comes from the experiment. In this introduction to probability, we make use of the basic