Page 38 - Applied statistics and probability for engineers

16 Chapter 2/Probability

Learning Objectives
After careful study of this chapter, you should be able to do the following:

1. Understand and describe sample spaces and events for random experiments with graphs, tables, lists,
or tree diagrams
2. Interpret probabilities and use the probabilities of outcomes to calculate probabilities of events in
discrete sample spaces
3. Use permuations and combinations to count the number of outcomes in both an event and the
sample space
4. Calculate the probabilities of joint events such as unions and intersections from the probabilities of
individual events
5. Interpret and calculate conditional probabilities of events
6. Determine the independence of events and use independence to calculate probabilities
7. Use Bayes’ theorem to calculate conditional probabilities
8. Understand random variables

2-1 Sample Spaces and Events

2-1.1 RANDOM EXPERIMENTS

If we measure the current in a thin copper wire, we are conducting an experiment. However,
day-to-day repetitions of the measurement can differ slightly because of small variations in
variables that are not controlled in our experiment, including changes in ambient temperatures,
slight variations in the gauge and small impurities in the chemical composition of the wire
(if different locations are selected), and current source drifts. Consequently, this experiment
(as well as many we conduct) is said to have a random component. In some cases, the ran-
dom variations are small enough, relative to our experimental goals, that they can be ignored.
However, no matter how carefully our experiment is designed and conducted, the variation is
almost always present, and its magnitude can be large enough that the important conclusions
from our experiment are not obvious. In these cases, the methods presented in this book for
modeling and analyzing experimental results are quite valuable.
Our goal is to understand, quantify, and model the type of variations that we often
encounter. When we incorporate the variation into our thinking and analyses, we can make
informed judgments from our results that are not invalidated by the variation.
Models and analyses that include variation are not different from models used in other
areas of engineering and science. Fig. 2-1 displays the important components. A math-
ematical model (or abstraction) of the physical system is developed. It need not be a per-
fect abstraction. For example, Newton’s laws are not perfect descriptions of our physical
universe. Still, they are useful models that can be studied and analyzed to approximately
quantify the performance of a wide range of engineered products. Given a mathematical
abstraction that is validated with measurements from our system, we can use the model to
understand, describe, and quantify important aspects of the physical system and predict the
response of the system to inputs.
Throughout this text, we discuss models that allow for variations in the outputs of a sys-
tem, even though the variables that we control are not purposely changed during our study.
Fig. 2-2 graphically displays a model that incorporates uncontrollable inputs (noise) that
combine with the controllable inputs to produce the output of our system. Because of the
uncontrollable inputs, the same settings for the controllable inputs do not result in identical
outputs every time the system is measured.