STATISTICS IN SUMMARY
Chapter Specifics
• A probability model describes a random phenomenon by telling what outcomes are possible and how to assign probabilities to them.
• There are two simple ways to give a probability model. The first assigns a probability to each individual outcome. These probabilities must be numbers between 0 and 1 (Rule A), and they must add to exactly 1 (Rule B). To find the probability of any event, add the probabilities of the outcomes that make up the event.
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• The second kind of probability model assigns probabilities as areas under a density curve, such as a Normal curve. The total probability is 1 because the total area under the curve is 1. This kind of probability model is often used to describe the sampling distribution of a statistic. This is the pattern of values of the statistic in many samples from the same population.
• All legitimate assignments of probability, whether data based or personal, obey the same probability rules. So the mathematics of probability is always the same.
• Odds of Y to Z that an event occurs corresponds to a probability of Z/(Y + Z ).
This chapter continues the discussion of probability that we began in Chapter 17. Here, we examine the formal mathematics of probability, embodied in probability models and probability rules. Probability models and rules provide the tools for describing and predicting the long-run behavior of random phenomena.
In this chapter, we also begin to formalize the process, first mentioned in Chapter 3, of using a statistic to estimate an unknown parameter. In particular, the sampling distribution will be the “probabilistic” tool we use to generalize from data produced by random samples and randomized comparative experiments to some wider population. Exactly how we do this will be the subject of Part IV.
CASE STUDY EVALUATED Look again at Table 18.1, discussed in the Case Study that opened the chapter.
1. Do the probabilities in this table follow the rules given on page 429?
2. On some websites, the probabilities are given as betting odds of winning. If you are a bookie setting betting odds, should you set the odds so the probabilities sum to greater than 1, exactly 1, or less than 1? Discuss.
Online Resources
• The Snapshots video Probability introduces the concepts of randomness, probability, and probability models in the context of some examples.
• The StatBoards video The Four Basic Probability Rules discusses the basic probability rules in the context of some simple examples.