Florida State University College of Law Research Center Statistics Concepts for Law Students Fall ‘08 Workshop Jon R. Lutz

Florida State University College of Law Research Center Coin Flip Odds when you flip a coinOdds when you flip a coin 50/50 or ½ that it will be heads or tails50/50 or ½ that it will be heads or tails Each flip of the coin has the same oddsEach flip of the coin has the same odds IndependenceIndependence What are the odds of two heads in a rowWhat are the odds of two heads in a row Multiply the oddsMultiply the odds ½ x ½ = ¼½ x ½ = ¼

Florida State University College of Law Research Center Consider the two patterns HTH and HTT. Which of the following is true? The average number of tosses until head tail head is larger than the average number of tosses until head tail tailThe 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 tailThe 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.The average number of tosses until head tail head is smaller than the average number of tosses until head tail tail.

Florida State University College of Law Research Center Formula 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.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.

Florida State University College of Law Research Center Epidemiology There is a test for a disease that is 99% accurateThere 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 diseaseYou test one person randomly – They test positive – What are the odds they have the disease

Florida State University College of Law Research Center Rarity of the disease Let’s say the disease is pretty rare, that only 1 in 10,000 people have the diseaseLet’s say the disease is pretty rare, that only 1 in 10,000 people have the disease

Florida State University College of Law Research Center Rarity 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.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.

Florida State University College of Law Research Center Rarity 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.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.

Florida State University College of Law Research Center Rarity What are the odds that the one individual who was initially tested, actually has the disease?What are the odds that the one individual who was initially tested, actually has the disease? Considerably less than 1%.Considerably less than 1%.

Florida State University College of Law Research Center Sally Clark Clark’s first son died a few weeks after birthClark’s first son died a few weeks after birth Clark’s second son also died shortly after birthClark’s second son also died shortly after birth Both cases appeared to be from Sudden Infant Death SyndromeBoth cases appeared to be from Sudden Infant Death Syndrome

Florida State University College of Law Research Center Pediatrician A pediatrician testified that two children dying from SIDS from and educated and affluent family (which the Clark family was) was very unlikelyA pediatrician testified that two children dying from SIDS from and educated and affluent family (which the Clark family was) was very unlikely

Florida State University College of Law Research Center Pediatrician SIDS occurs in non-smoking healthy household at a rate of about 1/8500

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

Florida State University College of Law Research Center Conviction Sally Clark was convictedSally Clark was convicted

Florida State University College of Law Research Center Problems Are there any statistical problems in this analyis?Are there any statistical problems in this analyis?

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

Florida State University College of Law Research Center Second Problem No evidence of independence was presented at the trial.No evidence of independence was presented at the trial. There are both genetic and environmental conditions that increase the risk of SIDSThere are both genetic and environmental conditions that increase the risk of SIDS

Florida State University College of Law Research Center Staph Infection 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.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.)(Somehow at the original trial the prosecution was aware of this but the defense was not.)

Florida State University College of Law Research Center Released After three years in prison her second appeal was successful and she was released.After three years in prison her second appeal was successful and she was released.

Florida State University College of Law Research Center Ending Unfortunately this does not have a happy endingUnfortunately this does not have a happy ending

Florida State University College of Law Research Center Sources How juries get fooled by statisticsHow juries get fooled by statistics &q=Common+Statistical+Misunderstandings&total=2&star t=0&num=10&so=0&type=search&plindex= &q=Common+Statistical+Misunderstandings&total=2&star t=0&num=10&so=0&type=search&plindex=0 The Bench Bench Federal Judicial Center c_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256C A30068A597Federal Judicial Center c_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256C A30068A597 c_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256C A30068A597 c_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256C A30068A597

Florida State University College of Law Research Center Sources Jury Understanding of DNA Evidence: An Empirical Assessment of Presentation Formats for Trace Evidence with a Relatively Small Random Match Probability Understanding of DNA Evidence: An Empirical Assessment of Presentation Formats for Trace Evidence with a Relatively Small Random Match Probability The Use of DNA statistics in criminal trials 46H1XKM- C&_user= &_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=sea rch&_sort=d&view=c&_version=1&_urlVersion=0&_userid= &md5= ae6342fa95b322c3af3ea07The Use of DNA statistics in criminal trials 46H1XKM- C&_user= &_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=sea rch&_sort=d&view=c&_version=1&_urlVersion=0&_userid= &md5= ae6342fa95b322c3af3ea H1XKM- C&_user= &_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=sea rch&_sort=d&view=c&_version=1&_urlVersion=0&_userid= &md5= ae6342fa95b322c3af3ea H1XKM- C&_user= &_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=sea rch&_sort=d&view=c&_version=1&_urlVersion=0&_userid= &md5= ae6342fa95b322c3af3ea07

Florida State University College of Law Research Center The End