CHAPTER 4

  1. 1. An informative and entertaining account of the origins of probability theory is Florence N. David, Games, Gods and Gambling, Charles Griffin, London, 1962.

  2. 2. See dupont.com/.

  3. 3. You can find a mathematical explanation of Benford’s law in Ted Hill, “The first-digit phenomenon,’’ American Scientist, 86 (1996), pp. 358–363; and Ted Hill, “The difficulty of faking data,’’ Chance, 12, No. 3 (1999), pp. 27–31. Applications in fraud detection are discussed in the second paper by Hill and in Mark A. Nigrini, “I’ve got your number,’’ Journal of Accountancy, May 1999, available online at www.journalofaccountancy.com/issues/1999/may/nigrini.html.

  4. 4. Royal Statistical Society news release, “Royal Statistical Society concerned by issues raised in Sally Clark case,’’ October 23, 2001, at www.rss.org.uk. For background, see an editorial and article in The Economist, January 22, 2004. The editorial is entitled “The probability of injustice.’’

  5. 5. See cdc.gov/mmwr/preview/mmwrhtml/mm57e618a1.htm.

  6. 6. See the Note 5.

  7. 7. See bloodbook.com/world-abo.html for the distribution of blood types for various groups of people.

  8. 8. From Statistics Canada, www.statcan.ca.

  9. 9. We use both for the random variable, which takes different values in repeated sampling, and for the numerical value of the random variable in a particular sample. Similarly, and stand both for random variables and for specific values. This notation is mathematically imprecise but statistically convenient.

  10. 10. We will consider only the case in which takes a finite number of possible values. The same ideas, implemented with more advanced mathematics, apply to random variables with an infinite but still countable collection of values.

    N-5

  11. 11. Based on a Pew Internet report, “Teens and distracted driving,’’ available from pewinternet.org/Reports/2009/Teens-and-Distracted-Driving.aspx.

  12. 12. See pewinternet.org/Reports/2009/17-Twitter-and-Status-Updating-Fall-2009.aspx.

  13. 13. The mean of a continuous random variable with density function can be found by integration:

    This integral is a kind of weighted average, analogous to the discrete-case mean

    The variance of a continuous random variable is the average squared deviation of the values of from their mean, found by the integral

  14. 14. See A. Tversky and D. Kahneman, “Belief in the law of small numbers,’’ Psychological Bulletin, 76 (1971), pp. 105–110, and other writings of these authors for a full account of our misperception of randomness.

  15. 15. Probabilities involving runs can be quite difficult to compute. That the probability of a run of three or more heads in 10 independent tosses of a fair coin is (1/2) + (1/128) = 0.508 can be found by clever counting. A general treatment using advanced methods appears in Section XIII.7 of William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, 1968.

  16. 16. R. Vallone and A. Tversky, “The hot hand in basketball: On the misperception of random sequences,’’ Cognitive Psychology, 17 (1985), pp. 295–314. A later series of articles that debate the independence question is A. Tversky and T. Gilovich, “The cold facts about the ‘hot hand’ in basketball,’’ Chance, 2, No. 1 (1989), pp. 16–21; P. D. Larkey, R. A. Smith, and J. B. Kadane, “It’s OK to believe in the ‘hot hand,’ ’’ Chance, 2, No. 4 (1989), pp. 22–30; and A. Tversky and T. Gilovich, “The ‘hot hand’: Statistical reality or cognitive illusion?’’ Chance, 2, No. 4 (1989), pp. 31–34.

  17. 17. Based on a study discussed in S. Atkinson, G. McCabe, C. Weaver, S. Abrams, and K. O’Brien, “Are current calcium recommendations for adolescents higher than needed to achieve optimal peak bone mass? The controversy,’’ Journal of Nutrition, 138, No. 6 (2008), pp. 1182–1186.

  18. 18. Based on a study described in Corby C. Martin et al., “Children in school cafeterias select foods containing more saturated fat and energy than the Institute of Medicine recommendations,’’ Journal of Nutrition, 140 (2010), pp. 1653–1660.

  19. 19. Based on The Ethics of American Youth–-2012, available from the Josephson Institute, charactercounts.org/wp-content/uploads/2014/02/ReportCard-2012-DataTables.pdf.

  20. 20. See nces.ed.gov/programs/digest. Data are from the 2012 Digest of Education Statistics.