florida state university college of law research center statistics concepts for law students fall...

Florida State University College of Law Research Center

StatisticsStatistics

Concepts for Law Students

Fall ‘08 WorkshopJon R. Lutz


Coin FlipCoin Flip

• Odds when you flip a coinOdds when you flip a coin• 50/50 or ½ that it will be heads or tails 50/50 or ½ that it will be heads or tails

• Each flip of the coin has the same odds Each 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


Consider the two patterns HTH and Consider the two patterns HTH and HTT. Which of the following is true? HTT. Which of the following is true?

• The average number of tosses until head tail The average number of tosses until head tail head is larger than the average number of head is larger than the average number of tosses until head tail tail tosses until head tail tail

• The average number of tosses until head tail The average number of tosses until head tail head is the same as the average number of head is the same as the average number of tosses until head tail tail tosses until head tail tail

• The average number of tosses until head tail The average number of tosses until head tail head is smaller than the average number of head is smaller than the average number of tosses until head tail tail. tosses until head tail tail.


Formula Formula

• To get the answer, look at the n "end" subsequences (X_n), (X_{n-1}, 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_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_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 (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 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 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.such k to get the expected number of tosses you're looking for.

Let's look at the two examples discussed: HTH and HTT. 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 For HTH, we have that H = H, TH != HT, HTH = HTH, so we get 2 + 2^3 = 10.= 10.

For HTT, we have that T != H, TT != HT, HTT = HTT, so we get 2^3 = 8.For HTT, we have that T != H, TT != HT, HTT = HTT, so we get 2^3 = 8.


EpidemiologyEpidemiology

• There is a test for a disease that is 99% There is a test for a disease that is 99% accurateaccurate• You test one person randomly – They test positive You test one person randomly – They test positive

– What are the odds they have the disease– What are the odds they have the disease


Rarity of the diseaseRarity of the disease

• Let’s say the disease is pretty rare, that only 1 Let’s say the disease is pretty rare, that only 1 in 10,000 people have the diseasein 10,000 people have the disease


RarityRarity

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


RarityRarity

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


RarityRarity

• What are the odds that the one individual who What are the odds that the one individual who was initially tested, actually has the disease? was initially tested, actually has the disease?

• Considerably less than 1%. Considerably less than 1%.


Sally Clark 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 birth Clark’s second son also died shortly after birth • Both cases appeared to be from Sudden Infant Both cases appeared to be from Sudden Infant

Death SyndromeDeath Syndrome


PediatricianPediatrician

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


Pediatrician Pediatrician

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


Pediatrician Pediatrician

• That two SIDS death would occur at a rate of That two SIDS death would occur at a rate of 1/8500 X 1/8500 or about 1 in 73, 000,000 1/8500 X 1/8500 or about 1 in 73, 000,000 births. births.

• It was presented to the jury that the chances It was presented to the jury that the chances that she was innocent was 1 in 73,000,000that she was innocent was 1 in 73,000,000


ConvictionConviction

• Sally Clark was convictedSally Clark was convicted


ProblemsProblems

• Are there any statistical problems in this Are there any statistical problems in this analyis?analyis?


First Problem First Problem

• One problem is that the relative rareness of One problem is that the relative rareness of parents who kill their children was not parents who kill their children was not presented. presented. • This is essentially the same problem we saw with This is essentially the same problem we saw with

the epidemiological problem the test for a disease the epidemiological problem the test for a disease that is 99% accurate. that is 99% accurate.


Second ProblemSecond Problem

• No evidence of independence was presented at No evidence of independence was presented at the trial. the trial.

• There are both genetic and environmental There are both genetic and environmental conditions that increase the risk of SIDSconditions that increase the risk of SIDS


Staph InfectionStaph Infection

• In this case it was revealed on appeal that one In this case it was revealed on appeal that one of the children had a staph infection at the time of the children had a staph infection at the time of death which was probably contributory to of death which was probably contributory to the death. the death. • (Somehow at the original trial the prosecution was (Somehow at the original trial the prosecution was

aware of this but the defense was not.)aware of this but the defense was not.)


ReleasedReleased

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


EndingEnding

• Unfortunately this does not have a happy Unfortunately this does not have a happy endingending


SourcesSources

• How juries get fooled by statisticsHow 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 The Bench http://www.unr.edu/bench/

• Federal Judicial Center http://www.fjc.gov/library/fjc_catalog.nsf/autoframepage!Federal Judicial Center http://www.fjc.gov/library/fjc_catalog.nsf/autoframepage!openform&url=/library/fjc_catalog.nsf/DPublication!openform&url=/library/fjc_catalog.nsf/DPublication!openform&parentunid=42E5BCA62EA1E03B85256CA30068A597openform&parentunid=42E5BCA62EA1E03B85256CA30068A597


SourcesSources

• Jury Understanding of DNA Evidence: An Empirical Assessment of Presentation Jury Understanding of DNA Evidence: An Empirical Assessment of Presentation Formats for Trace Evidence with a Relatively Small Random Match Probability Formats for Trace Evidence with a Relatively Small Random Match Probability http://papers.ssrn.com/sol3/papers.cfm?abstract_id=462880 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=462880

• The Use of DNA statistics in criminal trials http://www.sciencedirect.com/science?The Use of DNA statistics in criminal trials http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6T6W-46H1XKM-_ob=ArticleURL&_udi=B6T6W-46H1XKM-C&_user=2139768&_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=seaC&_user=2139768&_coverDate=08%2F28%2F2002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=2139768&md5=4682rch&_sort=d&view=c&_version=1&_urlVersion=0&_userid=2139768&md5=468204718ae6342fa95b322c3af3ea0704718ae6342fa95b322c3af3ea07

• http://en.wikipedia.org/wiki/Sally_Clarkhttp://en.wikipedia.org/wiki/Sally_Clark


The End

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