# Chapter 4PROBABILITY AND RELATED CONCEPTS

As indicated in Chapter 1, a fundamental goal is making inferences about a population of individuals or things based on a sample or subset of the population that is available. Basic probability plays a key role when addressing this issue. This chapter covers the fundamentals of probability and some related concepts and results that will be needed. Section 4.7 is particularly important in terms of understanding one of the reasons that many of the statistical methods developed during the last 50 years have practical value.

## 4.1 The Meaning of Probability

Coming up with an acceptable definition of the term probability turns out to be a nontrivial task. For example, imagine a bowl with 100 marbles, of which 50 are blue and the other 50 are green. When someone picks a marble from the bowl, without looking, what is the probability that the marble will be blue? A seemingly natural response is 0.5 because half the marbles are blue. But suppose all of the green marbles are on the bottom of the bowl and the person picking a marble has a penchant for picking marbles from those on top. Then, the probability of a blue marble is virtually 1; it will happen with near certainty unless this individual decides to reach deeply into the bowl. Of course, we can mix the marbles in the bowl so that not all of the blue marbles are at the top. But when is the bowl of marbles sufficiently mixed so that the probability of a blue marble is 0.5? A response might ...