Editing Base rate fallacy (section)

==Examples in law==
{{See also|People v. Collins}}

===O. J. Simpson trial===
{{see also|O. J. Simpson murder trial}}

[[O. J. Simpson]] was tried and acquitted in 1995 for the murders of his ex-wife Nicole Brown Simpson and her friend Ronald Goldman.

Crime scene blood matched Simpson's with characteristics shared by 1 in 400 people. However, the defense argued that the number of people from Los Angeles matching the sample could fill a football stadium and that the figure of 1 in 400 was useless.<ref>Robertson, B., & Vignaux, G. A. (1995). '' Interpreting evidence: Evaluating forensic evidence in the courtroom. '' Chichester: John Wiley and Sons.</ref><ref>Rossmo, D. Kim (2009). '' Criminal Investigative Failures. '' CRC Press Taylor & Francis Group.</ref> It would have been incorrect, and an example of prosecutor's fallacy, to rely solely on the "1 in 400" figure to deduce that a given person matching the sample would be likely to be the culprit.

[[File:OJ Simpson case frequency tree.svg|thumb|Frequency tree of 100 000 battered American women showing the base rate fallacy made by the defense in the [[O. J. Simpson murder trial]]]]

In the same trial, the prosecution presented evidence that Simpson had been violent toward his wife. The defense argued that there was only one woman murdered for every 2500 women who were subjected to spousal abuse, and that any history of Simpson being violent toward his wife was irrelevant to the trial. However, the reasoning behind the defense's calculation was fallacious. According to author [[Gerd Gigerenzer]], the correct probability requires additional context: Simpson's wife had not only been subjected to domestic violence, but rather subjected to domestic violence (by Simpson) {{em|and}} killed (by someone). Gigerenzer writes "the chances that a batterer actually murdered his partner, given that she has been killed, is about 8 in 9 or approximately 90%".<ref>Gigerenzer, G., Reckoning with Risk: Learning to Live with Uncertainty, Penguin, (2003)</ref> While most cases of spousal abuse do not end in murder, most cases of murder where there is a history of spousal abuse were committed by the spouse.

===Sally Clark case===
[[Sally Clark]], a British woman, was accused in 1998 of having killed her first child at 11 weeks of age and then her second child at 8 weeks of age. The prosecution had [[expert witness]] Sir [[Roy Meadow]], a professor and consultant paediatrician,<ref>{{Cite magazine |date=30 November 1998 |title=Resolution adopted by the Senate (21 October 1998) on the retirement of Professor Sir Roy Meadow |url=http://reporter.leeds.ac.uk/428/mead.htm |url-status=dead |archive-url=https://web.archive.org/web/20160416125642/http://reporter.leeds.ac.uk/428/mead.htm |archive-date=2016-04-16 |access-date=2015-10-17 |magazine=Reporter |publisher=[[University of Leeds]] |number=428}}</ref> testify that the probability of two children in the same family dying from [[SIDS]] is about 1 in 73 million. That was much less frequent than the actual rate measured in [[Epidemiology|historical data]]{{snd}} Meadow estimated it from single-SIDS death data, and the assumption that the probability of such deaths should be [[uncorrelated]] between infants.<ref>The population-wide probability of a SIDS fatality was about 1 in 1,303; Meadow generated his 1-in-73 million estimate from the lesser probability of SIDS death in the Clark household, which had lower risk factors (e.g. non-smoking). In this sub-population he estimated the probability of a single death at 1 in 8,500. See: {{cite web |last=Joyce |first=H. |date=September 2002 |title=Beyond reasonable doubt |url=http://plus.maths.org/issue21/features/clark/ |access-date=2010-06-12 |publisher=plus.maths.org |format=pdf}}. Professor Ray Hill questioned even this first step (1/8,500 vs 1/1,300) in two ways: firstly, on the grounds that it was [[bias]]ed, excluding those factors that increased risk (especially that both children were boys) and (more importantly) because reductions in SIDS risk factors will proportionately reduce murder risk factors, so that the relative frequencies of [[Münchausen syndrome by proxy]] and SIDS will remain in the same ratio as in the general population: {{cite web |last=Hill |first=Ray |year=2002 |title=Cot Death or Murder? – Weighing the Probabilities |url=http://www.mrbartonmaths.com/resources/a%20level/s1/Beyond%20reasonable%20doubt.doc |quote=it is patently unfair to use the characteristics which basically make her a good, clean-living, mother as factors which count against her.  Yes, we can agree that such factors make a natural death less likely – but those same characteristics also make murder less likely.}}
</ref>

Meadow acknowledged that 1-in-73&nbsp;million is not an impossibility, but argued that such accidents would happen "once every hundred years" and that, in a country of 15 million 2-child families, it is vastly more likely that the double-deaths are due to [[Münchausen syndrome by proxy]] than to such a rare accident. However, there is good reason to suppose that the likelihood of a death from SIDS in a family is significantly greater if a previous child has already died in these circumstances, (a [[genetic predisposition]] to SIDS is likely to invalidate that assumed [[statistical independence]]<ref>{{Cite news |last1=Sweeney |first1=John |last2=Law |first2=Bill |date=July 15, 2001 |title=Gene find casts doubt on double 'cot death' murders |url=http://observer.guardian.co.uk/print/0,,4221973-102285,00.html |archive-url=https://archive.today/20120711221623/http://observer.guardian.co.uk/print/0,,4221973-102285,00.html |archive-date=2012-07-11 |work=[[The Observer]]}}</ref>) making some families more susceptible to SIDS and the error an outcome of the [[ecological fallacy]].<ref>{{cite web |author=Vincent Scheurer |title=Convicted on Statistics? |url=http://understandinguncertainty.org/node/545#notes |access-date=2010-05-21}}</ref> The likelihood of two SIDS deaths in the same family cannot be soundly [[Estimation theory|estimated]] by squaring the likelihood of a single such death in all otherwise similar families.<ref>{{cite journal |last=Hill |first=R. |year=2004 |title=Multiple sudden infant deaths – coincidence or beyond coincidence? |url=http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf |url-status=dead |journal=Paediatric and Perinatal Epidemiology |volume=18 |issue=5 |page=321 |doi=10.1111/j.1365-3016.2004.00560.x |pmid=15367318 |archive-url=https://web.archive.org/web/20120830145646/http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf |archive-date=2012-08-30 |access-date=2010-06-13}}</ref>

The 1-in-73&nbsp;million figure greatly underestimated the chance of two successive accidents, but even if that assessment were accurate, the court seems to have missed the fact that the 1-in-73&nbsp;million number meant nothing on its own. As an ''[[a priori]]'' probability, it should have been weighed against the ''a priori'' probabilities of the alternatives. Given that two deaths had occurred, one of the following explanations must be true, and all of them are ''a priori'' extremely improbable:
# Two successive deaths in the same family, both by SIDS
# Double homicide (the prosecution's case)
# Other possibilities (including one homicide and one case of SIDS)

It is unclear whether an estimate of the probability for the second possibility was ever proposed during the trial, or whether the comparison of the first two probabilities was understood to be the key estimate to make in the statistical analysis assessing the prosecution's case against the case for innocence.

Clark was convicted in 1999, resulting in a press release by the [[Royal Statistical Society]] which pointed out the mistakes.<ref>{{cite web |date=23 October 2001 |title=Royal Statistical Society concerned by issues raised in Sally Clark case |url=http://www.rss.org.uk/uploadedfiles/documentlibrary/744.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110824151124/http://www.rss.org.uk/uploadedfiles/documentlibrary/744.pdf |archive-date=24 August 2011 |quote=Society does not tolerate doctors making serious clinical errors because it is widely understood that such errors could mean the difference between life and death. The case of R v. Sally Clark is one example of a medical expert witness making a serious statistical error, one which may have had a profound effect on the outcome of the case}}</ref>

In 2002, Ray Hill (a mathematics professor at [[University of Salford|Salford]]) attempted to accurately compare the chances of these two possible explanations; he concluded that successive accidents are between 4.5 and 9 times more likely than are successive murders, so that the ''a priori'' [[odds]] of Clark's guilt were between 4.5 to 1 and 9 to 1 against.<ref>The uncertainty in this range is mainly driven by uncertainty in the likelihood of killing a second child, having killed a first, see: {{cite journal |last=Hill |first=R. |year=2004 |title=Multiple sudden infant deaths – coincidence or beyond coincidence? |url=http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf |url-status=dead |journal=Paediatric and Perinatal Epidemiology |volume=18 |issue=5 |pages=322–323 |doi=10.1111/j.1365-3016.2004.00560.x |pmid=15367318 |archive-url=https://web.archive.org/web/20120830145646/http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf |archive-date=2012-08-30 |access-date=2010-06-13}}</ref>

After the court found that the forensic pathologist who had examined both babies had withheld [[exculpatory evidence]], a higher court later quashed Clark's conviction, on 29 January 2003.<ref>{{cite web |title=R v Clark. [2003] EWCA Crim 1020 (11 April 2003) |url=http://www.bailii.org/ew/cases/EWCA/Crim/2003/1020.html |website=www.bailii.org}}</ref>