STATISTICS IN SUMMARY
Chapter Specifics
• Some things in the world, both natural and of human design, are random. That is, their outcomes have a clear pattern in very many repetitions even though the outcome of any one trial is unpredictable.
• Probability describes the long-term regularity of random phenomena. The probability of an outcome is the proportion of very many repetitions on which that outcome occurs. A probability is a number between 0 (the outcome never occurs) and 1 (always occurs). We emphasize this kind of probability because it is based on data.
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• Probabilities describe only what happens in the long run. Short runs of random phenomena like tossing coins or shooting a basketball often don’t look random to us because they do not show the regularity that in fact emerges only in very many repetitions.
• Personal probabilities express an individual’s personal judgment of how likely outcomes are. Personal probabilities are also numbers between 0 and 1. Different people can have different personal probabilities, and a personal probability need not agree with a proportion based on data about similar cases.
This chapter begins our study of the mathematics of chance or “probability.” The important fact is that random phenomena are unpredictable in the short run but have a regular and predictable behavior in the long run.
The long-run behavior of random phenomena will help us understand both why and in what way we can trust random samples and randomized comparative experiments, the subjects of Chapters 2 through 6. It is the key to generalizing what we learn from data produced by random samples and randomized comparative experiments to some wider universe or population. We will study how this is done in Part IV. As a first step in this direction, we will look more carefully at the basic rules of probability in the next chapter.
CASE STUDY EVALUATED In the Case Study described at the beginning of this chapter, you were told that if birth dates are random and independent, the chance that three children, selected at random, are all born on Leap Day is about 1 in 3 billion.
1. Go to the most recent Statistical Abstract (online at http://www.census.gov
/library/publications/time-series/statistical_abstracts.html) and look under Population, Households and Families, Families by Number of Own Children under 18 Years Old. How many families in the United States have at least three children under 18 years old?2. For the time being, assume that the families you found in the previous question all have exactly three children. Explain why the probability that some family in the United States has three children all born on Leap Day is much larger than 1 in 3 billion.
3. Next, consider the fact that not all the families have exactly three children, that the number of families in Question 1 does not include those with children over the age of 18, and that parents might intentionally try to conceive children with the same birth date (in fact, the Associated Press news article mentioned that after they had a child born on Leap Day, the couple in Utah tried to have a child on subsequent Leap Days). Write a paragraph discussing whether the “surprising” coincidence described in the Case Study that began this chapter is as surprising as it might first appear.
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Online Resources
• The first half of the Snapshots video Probability introduces the concepts of randomness and probability in the context of weather forecasts.
• The StatBoards video Myths about Chance Behavior discusses some common misconceptions about chancer behavior such as short-run regularity, surprising coincidences, and the law of averages.