DOCSLIB.ORG

Probability, Statistics, and Justice

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Probability, Statistics, and Justice Probability, Statistics, and Justice Jeffrey S. Rosenthal Professor Department of Statistics University of Toronto jeff@math.toronto.edu www.probability.ca (Canadian Institute for the Administration of Justice, Annual Conference, October 10, 2013) (1/45) I'm a Professor of Statistics. A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . (2/45) A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . I'm a Professor of Statistics. (2/45) Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) About Me . I'm a Professor of Statistics. A typical day's work: (2/45) About Me . I'm a Professor of Statistics. A typical day's work: Therefore p − (d) + (d) α (βi ) = α (βi − `= d) 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) d + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) ≈ α (βi )− p φ − − 2 2 1=2 (d) d 6I (βi ) (d) 1=2 (d) 0 2 (d) 000 (`(βi )I (βi )) −` − exp(−` (β )K (βi )=6) p × i 2 d I 1=2(β(d))` `K 000(β(d)) ×φ − i + i 2 1=2 (d) 6I (βi ) 1=2 1=2 d d 0 Then, since ` ≈ ` + `0 ≈ ` + `0 = ` + `` , we have that d1=2 0 d1=2 1 h `` µ(β(d)) ≈ − α+` + ` + × i 2d 1=2 d 1=2 (d) 1=2 (d) 0 1=2 (d) 000 (d) + (d) (`(βi )I (βi )) −` I (βi )` `K (βi ) α (βi )− p φ − − − 2 2 1=2 (d) d 6I (βi ) (2/45) And then one day I wrote a successful book . (3/45) Opinion Polls . Then I was interviewed by the media about: (4/45) Then I was interviewed by the media about: Opinion Polls . (4/45) Crime statistics . (5/45) Pedestrian death counts... (6/45) Making decisions . (7/45) Game show strategies . (8/45) The Maple Leafs . To: [email protected] From: jeff@math.toronto.edu (Jeffrey Rosenthal) Date: Wed, 12 Apr 2006 16:11:08 -0400 (EDT) Subject: my calculations Hi Mike, good talking to you on the phone just now. I assumed that in each game, each team has probability 45% of getting two points, 10% of getting one point, and 45% of getting zero points. Those figures then lead to the following probabilities for Toronto to beat or tie each of the various other teams (in total points at end of season): Prob that Toronto beats Montreal = 0.17% Prob that Toronto ties Montreal = 0.30% Prob that Toronto ties Atlanta = 11.5% Prob that Toronto beats Atlanta = 30.2% Prob that Toronto ties Tampa Bay = 3.6% Prob that Toronto beats Tampa Bay = 2.1% * This gives a total probability of 5.8% (about one chance in 17) for Toronto to have a chance at the playoffs. (9/45) And lotteries . (10/45) . Lots of lotteries . (11/45) Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? (12/45) Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? (12/45) \Once is happenstance. Twice is coincidence. The third time it's enemy action." So what is the connection to JUSTICE? Auric Goldfinger: (12/45) So what is the connection to JUSTICE? Auric Goldfinger: \Once is happenstance. Twice is coincidence. The third time it's enemy action." (12/45) Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? (13/45) Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. (13/45) Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" (13/45) (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (13/45) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) (13/45) (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (13/45) and (b) illustrate why it must be used with caution. Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, (13/45) Probability, Statistics . and Justice? Statistics and Justice both involve evaluating evidence. Justice: \beyond a reasonable doubt" \balance of probabilities" \preponderance of the evidence" Statistics: \statistically significant” \the probability is more than X" \the p-value is less than Y" (“significant”?? \probability"?? \p-value"??) In this talk, I will try to: (a) explain how statistical reasoning works, and (b) illustrate why it must be used with caution. (13/45) Example: Can your friend distinguish Coke from Pepsi? Do a test! Guesses right the first time. Proof of ability? No, could be luck! What about twice in a row? three times? ten times? The p-value is the probability of such a result if it's just random. • Guess right once: p-value = 1=2 = 50%. • Guess right twice in a row: p-value = (1=2) × (1=2) = 25%. • Guess right five times in a row: multiply (\independent"): : p-value = (1=2) × (1=2) × (1=2) × (1=2) × (1=2) = 3:1%.
Load more

Recommended publications
  • Probability, Justice, and the Risk of Wrongful Conviction
    The Mathematics Enthusiast Volume 12 Number 1 Numbers 1, 2, & 3 Article 5 6-2015 Probability, Justice, and the Risk of Wrongful Conviction Jeffrey S. Rosenthal Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Rosenthal, Jeffrey S. (2015) "Probability, Justice, and the Risk of Wrongful Conviction," The Mathematics Enthusiast: Vol. 12 : No. 1 , Article 5. Available at: https://scholarworks.umt.edu/tme/vol12/iss1/5 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 12, no. 1,2&3, p. 11 Probability, Justice, and the Risk of Wrongful Conviction Jeffrey S. Rosenthal University of Toronto, Canada Abstract: We consider the issue of standards of proof in legal decisions from the point of view of probability. We compare ``balance of probabilities'' and ``beyond a reasonable doubt'' to the statistical use of p-values. We point out various fallacies which sometimes arise in legal reasoning. And we provide several examples of legal cases which involved probabilities, including some in which incorrect decisions were made and defendants were wrongfully convicted. Keywords: probability, statistics, p-value, balance of probabilities, beyond a reasonable doubt, standard of proof. Background and Context I am a professor of statistics, and most of my work is fairly technical and mathematical (see www.probability.ca/jeff/research.html).
    [Show full text]
  • Convictionbynumbers
    NEWS FEATURE NATURE|Vol 445|18 January 2007 CONVICTION BY NUMBERS Statistics have the power to trip everyone up — including judges and juries. Even when extra care is taken to get the numbers right in court, confusion often reigns. Mark Buchanan reports. n March 2003, when nurse Lucia de Berk A court of appeal later upheld de Berk’s con- rocket science, so a statistical expert was not faced trial in a Dutch court for charges of viction, but an advisory judicial committee needed.” murder and attempted murder, the statisti- has now been appointed by the central office The appeals court confirmed Clark’s origi- Ical evidence against her seemed compel- of public prosecutors to reassess the case. The nal conviction, but Dawid says that the written ling. Investigators had identified a number of committee’s decision is expected later this year, judgement showed that the jury had failed to ‘suspicious’ deaths and near deaths in hospital and it could recommend that the case be reo- grasp the relevant statistical issues. A second IMAGES.COM/CORBIS wards in which de Berk had worked from 1999 pened. But whatever the result, the case illus- appeal in 2003 freed Clark, concluding that to 2001, and records showed that she had been trates the ongoing difficulty of ensuring that the jury might have been misled in part by the present when many of those events took place. courts use statistical reasoning properly. “This statistics presented in the original trial. Without The statistical expert testifying in the case, is a serious problem,” says statistics professor a change in legal attitudes and procedures, math- Henk Elffers, reported that the chance that Philip Dawid at University College London, ematicians worry that statistical arguments will her presence was mere coincidence was only “but there is no easy solution.” continue to be misinterpreted by the courts.
    [Show full text]
  • “Your Honor, This Was Not a Coincidence!”
    Peter Grünwald March 2007 The Case of Lucia de Berk “Your Honor, this was not a coincidence!” • In 2004, the court of appeals convicted Lucia de Berk, a nurse from the Juliana Children’s Hospital in the Hague, to life On the (ab)use of statistics in the case against imprisonment for 7 murders and 3 Lucia de B. murder attempts. • De B. has been in prison for more than 5 Peter Grünwald years now, but has never confessed. CWI, Amsterdam • The case is now under review of the “Committee Posthumus-II” 2001 (september 16th) Juliana hospital notifies the police Menu –There were many more “incidents” during her shifts than during those of other nurses in her ward –Statistician calculates that the probability that something like this would happen by pure coincidence, is less than 1 in 342 million 1. The use of statistics –Trial gets (more than) substantial media attention 2003 Court convicts L. of 4 murders, 3 attempts • What the statistician did (evidence in first verdict) –Statistics is crucial part of the evidence • What went wrong (everything) –In appeal, experts for defense R. Meester (probabilist) and • The role of statistics in the final verdict M. van Lambalgen (logician) claim calculation is flawed 2004 Court of appeals convicts L. of 7 murders,3 attempts –Court says that this time, “statistics in the form of probability 2. Can we do better? calculations has not been used” –In fact, the court’s report has flawed statistics written all over it 2006 Prof. Ton Derksen writes a book about the case –Case goes to “Posthumus-II committee” –Richard Gill and myself write letter to the committee supporting Derksen’s statistical analysis Not a coincidence! - Evidence Seminar, University College London 1 Peter Grünwald March 2007 The Statistician’s Work 1.
    [Show full text]
  • Probability, Justice, and the Risk of Wrongful Conviction by Jeffrey S
    Probability, Justice, and the Risk of Wrongful Conviction by Jeffrey S. Rosenthal1 (September, 2014) 1 Background and Context I am a professor of statistics, and most of my work is fairly technical and mathematical2. But one day I wrote a book for the general public3, which did unexpected well, especially in Canada. I was then interviewed by the media about such diverse topics as lottery probabilities, public opinion polls, violent crime rates, sports statistics, and more, and was even involved in probing a major lottery retailer fraud scandal involving millions of dollars and criminal convictions4. I was also invited to give talks to all sorts of different groups, from insurance brokers to financial investors, from humour therapists to gambling addiction counselors. And then one day I was invited to speak to a prominent group of Cana- dian lawyers and judges. This invitation in turn led to others, and I ended up giving five different talks to five different groups of lawyers and judges (including the Irish Supreme Court and High Court justices) within a sin- gle year. This forced me to investigate the connection of probabilities and statistical analysis to the justice system, as I will now discuss. 2 Probability and Justice? What is the connection of probability and statistics to justice issues? Well, both topics involve evaluating evidence, i.e. determining whether the avail- able information is sufficient to draw certain conclusions. This perspective 1Professor, Department of Statistics, University of Toronto. Email: [email protected]. Web: www.probability.ca 2See: www.probability.ca/jeff/research.html 3Struck by Lightning: The Curious World of Probabilities, HarperCollins Canada 2005 and Joseph Henry Press 2006.
    [Show full text]