144 CHAPTER 4 Probability 4-3 Complements, Conditional Probability, and Bayes’ Theorem • Compute the probability of “at least one” occurrence of an event A. • Apply the multiplication rule by computing the probability of some event, given that some other event has already occurred. 4-4 Counting • Develop the ability to apply the fundamental counting rule, factorial rule, permutations rule, and combinations rule. • Distinguish between circumstances requiring the permutations rule and those requiring the combinations rule. 4-5 Simulations for Hypothesis Tests • Use simulations to determine when sample results are significantly low or significantly high, so that claims about population parameters can be tested. Key Concept Part 1 of this section includes basic concepts of probability. The single most important objective of this section is to learn how to interpret probability values, which are expressed as values between 0 and 1. A small probability, such as 0.001, corresponds to an event that rarely occurs. When interpreting probability values, it is also important to understand the rare event rule for inferential statistics, which is described later in Part 1 of this section. Part 2 of this section includes odds and how they relate to probabilities. Concepts related to odds are not needed for topics in the following chapters, but odds are commonly used in situations such as lotteries and gambling. 4-1 Basic Concepts of Probability PART 1 Basic Concepts of Probability Role of Probability in Statistics Probability plays a central role in the important statistical method of hypothesis testing introduced later in Chapter 8. Statisticians make decisions using data by rejecting explanations (such as chance) based on very low probabilities. See the following example illustrating the role of probability and a fundamental way that statisticians think. Researchers have made this claim (really, they have): Claim: “We have developed a gender selection method that greatly increases the likelihood of a baby being a girl.” Hypothesis Used When Testing the Preceding Claim: The method of gender selection has no effect, so that for couples using this method, about 50% of the births result in girls. CP EXAMPLE 1 Analyzing a Claim
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