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Statistics

Statistics. Concepts for Law Students . Fall ‘08 Workshop Jon R. Lutz. Odds when you flip a coin 50/50 or ½ that it will be heads or tails Each flip of the coin has the same odds Independence What are the odds of two heads in a row Multiply the odds ½ x ½ = ¼. Coin Flip.

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Statistics

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  1. Statistics Concepts for Law Students Fall ‘08 Workshop Jon R. Lutz

  2. Odds when you flip a coin 50/50 or ½ that it will be heads or tails Each flip of the coin has the same odds Independence What are the odds of two heads in a row Multiply the odds ½ x ½ = ¼ Coin Flip

  3. The average number of tosses until head tail head is larger than the average number of tosses until head tail tail The average number of tosses until head tail head is the same as the average number of tosses until head tail tail The average number of tosses until head tail head is smaller than the average number of tosses until head tail tail. Consider the two patterns HTH and HTT. Which of the following is true?

  4. To get the answer, look at the n "end" subsequences (X_n), (X_{n-1}, X_n), (X_{n-2}, X_{n-1}, X_n), ..., (X_{n-k}, X_{n-k+1}, ..., X_n), ...(X_1, ..., X_n), and compare them to the n "begin" subsequences (X_1), (X_1, X_2), (X_1,X_2,X_3), ..., (X_1, ..., X_k), ..., (X_1, ..., X_n). Count up the k's between 1 and n such that the "end" subsequence of length k is the same as the "begin" subsequence of length k, and sum up 2^k over all such k to get the expected number of tosses you're looking for.Let's look at the two examples discussed: HTH and HTT. For HTH, we have that H = H, TH != HT, HTH = HTH, so we get 2 + 2^3 = 10.For HTT, we have that T != H, TT != HT, HTT = HTT, so we get 2^3 = 8. Formula

  5. There is a test for a disease that is 99% accurate You test one person randomly – They test positive – What are the odds they have the disease Epidemiology

  6. Let’s say the disease is pretty rare, that only 1 in 10,000 people have the disease Rarity of the disease

  7. You test one million people. At that rarity 100 people would have the disease and since the test is 99% accurate, it would correctly identify 99 people as having the disease and it would miss one. Rarity

  8. You test one million people for the disease, since only about 100 will have the disease the overwhelming majority of the 1,000,000 people will not, but since the test is 99% accurate it will falsely identify 10,000 people as having the disease. Rarity

  9. What are the odds that the one individual who was initially tested, actually has the disease? Considerably less than 1%. Rarity

  10. Clark’s first son died a few weeks after birth Clark’s second son also died shortly after birth Both cases appeared to be from Sudden Infant Death Syndrome Sally Clark

  11. A pediatrician testified that two children dying from SIDS from and educated and affluent family (which the Clark family was) was very unlikely Pediatrician

  12. SIDS occurs in non-smoking healthy household at a rate of about 1/8500 Pediatrician

  13. That two SIDS death would occur at a rate of 1/8500 X 1/8500 or about 1 in 73, 000,000 births. It was presented to the jury that the chances that she was innocent was 1 in 73,000,000 Pediatrician

  14. Sally Clark was convicted Conviction

  15. Are there any statistical problems in this analyis? Problems

  16. One problem is that the relative rareness of parents who kill their children was not presented. This is essentially the same problem we saw with the epidemiological problem the test for a disease that is 99% accurate. First Problem

  17. No evidence of independence was presented at the trial. There are both genetic and environmental conditions that increase the risk of SIDS Second Problem

  18. In this case it was revealed on appeal that one of the children had a staph infection at the time of death which was probably contributory to the death. (Somehow at the original trial the prosecution was aware of this but the defense was not.) Staph Infection

  19. After three years in prison her second appeal was successful and she was released. Released

  20. Ending • Unfortunately this does not have a happy ending

  21. How juries get fooled by statistics http://video.google.com/videoplay?docid=-8018179996887789254&q=Common+Statistical+Misunderstandings&total=2&start=0&num=10&so=0&type=search&plindex=0 The Bench http://www.unr.edu/bench/ Federal Judicial Center http://www.fjc.gov/library/fjc_catalog.nsf/autoframepage!openform&url=/library/fjc_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256CA30068A597 Sources

  22. Jury Understanding of DNA Evidence: An Empirical Assessment of Presentation Formats for Trace Evidence with a Relatively Small Random Match Probability http://papers.ssrn.com/sol3/papers.cfm?abstract_id=462880 The Use of DNA statistics in criminal trials http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T6W-46H1XKM-C&_user=2139768&_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=2139768&md5=468204718ae6342fa95b322c3af3ea07 http://en.wikipedia.org/wiki/Sally_Clark Sources

  23. The End

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