Prosecutor's fallacy
The prosecutor's fallacy is a fallacy of statistical reasoning, typically used by the prosecution to argue for the guilt of a defendant during a criminal trial. Although it is named after prosecutors, it is not specific to them, and some variants of the fallacy can be used by defense lawyers arguing for the innocence of their client.
The following claim demonstrates the fallacy in the context of a prosecutor questioning an expert witness: “the odds of finding this evidence on an innocent man are so small that the jury can safely disregard the possibility that this defendant is innocent”.[1] The fallacy obscures that the odds of a defendant being innocent given said evidence in fact depends on the likely higher prior odds of the defendant being innocent, the explicitly lesser odds of the evidence in the case that he was innocent as mentioned, as well as the underlying cumulative odds of the evidence being on the defendant.
At its heart, the fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. For instance, if a perpetrator is known to have the same blood type as a defendant and 10% of the population share that blood type, then to argue on that basis alone that the probability of the defendant being guilty is 90% commits the prosecutor's fallacy (in a very simple form).
Concept
The terms "prosecutor's fallacy" and "defense attorney's fallacy" were originated by William C. Thompson and Edward Schumann in 1987.[2][3] The fallacy can arise from multiple testing, such as when evidence is compared against a large database. The size of the database elevates the likelihood of finding a match by pure chance alone; i.e., DNA evidence is soundest when a match is found after a single directed comparison because the existence of matches against a large database where the test sample is of poor quality may be less unlikely by mere chance.
The basic fallacy results from misunderstanding conditional probability and neglecting the prior odds of a defendant being guilty before that evidence was introduced. When a prosecutor has collected some evidence (for instance a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. If the DNA match is used to confirm guilt which is otherwise suspected then it is indeed strong evidence. However, if the DNA evidence is the sole evidence against the accused and the accused was picked out of a large database of DNA profiles, the odds of the match being made at random may be increased, and less damaging to the defendant. The odds in this scenario do not relate to the odds of being guilty, they relate to the odds of being picked at random. While the odds of being picked at random may be low for an individual condition implying guilt, i.e. a positive DNA match, the probability of being picked at random for any condition grows to 1 as more conditions are considered, as is the case in multiple testing. It is often the case that both innocence and guilt (i.e., accidental death and murder) are both highly improbable, though naturally one must be true, so the ratio of the likelihood of the "innocent scenario" to the "guilty scenario" is much more informative than the probability of the "guilty scenario" alone.
Examples
Conditional probability
In the fallacy of argument from rarity, an explanation for an observed event is said to be unlikely because the prior probability of that explanation is low. Consider this case: a lottery winner is accused of cheating, based on the improbability of winning. At the trial, the prosecutor calculates the (very small) probability of winning the lottery without cheating and argues that this is the chance of innocence. The logical flaw is that the prosecutor has failed to account for the large number of people who play the lottery. While the probability of any singular person winning is quite low, the probability of any person winning the lottery, given the number of people who play it, is very high.
In Berkson's paradox, conditional probability is mistaken for unconditional probability. This has led to several wrongful convictions of British mothers, accused of murdering two of their children in infancy, where the primary evidence against them was the statistical improbability of two children dying accidentally in the same household (under "Meadow's law"). Though multiple accidental (SIDS) deaths are rare, so are multiple murders; with only the facts of the deaths as evidence, it is the ratio of these (prior) improbabilities that gives the correct "posterior probability" of murder.[4]
Multiple testing
In another scenario, a crimescene DNA sample is compared against a database of 20,000 men. A match is found, that man is accused and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000. This does not mean the probability that the suspect is innocent is 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance.
Even if none of the men in the database left the crimescene DNA, a match by chance to an innocent is more likely than not. The chance of getting at least one match among the records is:
 ,
 where, explicitly:
 = probability of two DNA profiles matching by chance, after one check,
 = probability of not matching, after one check,
 = probably of not matching, after 20,000 checks, and
 = probably of matching, after 20,000 checks.
So, this evidence alone is an uncompelling data dredging result. If the culprit were in the database then he and one or more other men would probably be matched; in either case, it would be a fallacy to ignore the number of records searched when weighing the evidence. "Cold hits" like this on DNA databanks are now understood to require careful presentation as trial evidence.
Mathematical analysis
Finding a person innocent or guilty can be viewed in mathematical terms as a form of binary classification. If E is the observed evidence, and I stands for "accused is innocent" then consider the conditional probabilities:
 P(EI) is the probability that the "damning evidence" would be observed even when the accused is innocent (a "false positive").
 P(IE) is the probability that the accused is innocent, despite the evidence E.
With forensic evidence, P(EI) is tiny. The prosecutor wrongly concludes that P(IE) is comparatively tiny. (The Lucia de Berk prosecution is accused of exactly this error,[5] for example.) In fact, P(EI) and P(IE) are quite different; using Bayes' theorem:
where:
 P(I) is the probability of innocence independent of the test result (i.e. from all other evidence) and
 P(E) is the prior probability that the evidence would be observed (regardless of innocence).
This equation shows that a small does not imply a small in case of a large and a small . That is, if the accused is otherwise likely to be innocent and it is unlikely for anyone (guilty or innocent) to exhibit the observed evidence.
Note that
 P(E~I) is the probability that the evidence would identify a guilty suspect (not give a false negative). This is usually close to 100%, slightly increasing the inference of innocence over a test with false negatives. That inequality is concisely expressed in terms of odds:
The prosecutor is claiming a negligible chance of innocence, given the evidence, implying Odds(IE) > P(IE), or that:
A prosecutor conflating P(IE) with P(EI) makes a technical error whenever Odds(I) >> 1. This may be a harmless error if P(IE) is still negligible, but it is especially misleading otherwise (mistaking low statistical significance for high confidence).
Legal impact
Though the prosecutor's fallacy typically happens by mistake,[6] in the adversarial system lawyers are usually free to present statistical evidence as best suits their case; retrials are more commonly the result of the prosecutor's fallacy in expert witness testimony or in the judge's summation.[7]
Defense attorney's fallacy
Suppose there is a oneinamillion chance of a match given that the accused is innocent. The prosecutor says this means there is only a oneinamillion chance of innocence. But if everyone in a community of 10 million people is tested, one expects 10 matches even if all are innocent. The defense fallacy would be to reason that "10 matches were expected, so the accused is no more likely to be guilty than any of the other matches, thus the evidence suggests a 90% chance that the accused is innocent." and "As such, this evidence is irrelevant." The first part of the reasoning would be correct only in the case where there is no further evidence pointing to the defendant. On the second part, Thompson & Schumann wrote that the evidence should still be highly relevant because it "drastically narrows the group of people who are or could have been suspects, while failing to exclude the defendant" (page 171).[2][8]
Another way of saying this would be to point out that the defense attorney's calculation failed to take into account the prior probability of the defendant's guilt. If, for example, the police came up with a list of 10 suspects, all of whom had access to the crime scene, then it would be very illogical indeed to suggest that a test that offers a oneinamillion chance of a match would change the defendant's prior probability from 1 in 10 (10 percent) to 1 in a million (0.0001 percent). If nine innocent people were to be tested, the likelihood that the test would incorrectly match one (or more) of those people can be calculated as
 ,
or approximately 0.0009%. If, however, the other 9 suspects were tested and did not return a match, then the probability of the defendant's guilt has increased from the prior probability of 10% (1 in 10 suspects) to 99.9991% on the basis of the test. The defendant might argue that "lists of suspects compiled by police fail to include the guilty person in 50% of cases" — if that were true, then the defendant's guilt would have increased from the prior probability of 5% (50% of 10%) to 49.99955% on the basis of the test — in which case "reasonable doubt" could be claimed to exist despite the positive test result.
Possible examples of fallacious defense arguments
Authors have cited defense arguments in the O. J. Simpson murder trial as an example of this fallacy regarding the context in which the accused had been brought to court: crime scene blood matched Simpson's with characteristics shared by 1 in 400 people. The defense argued that a football stadium could be filled with Angelenos matching the sample and that the figure of 1 in 400 was useless.[9][10]
Also at the O. J. Simpson murder trial, the prosecution presented evidence that Simpson had been violent toward his wife, while 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 the context — that Simpson's wife had not only been subjected to domestic violence, but rather subjected to domestic violence (by Simpson) and murdered (by someone) — to be taken into account. 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%".[11] While most cases of spousal abuse do not end in murder, most cases of murder where there is a history of spousal abuse were murdered by their spouse.
The 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,[12] 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 historical data – Meadow estimated it from singleSIDS death data, and the assumption that the probability of such deaths should be uncorrelated between infants.[13]
Meadow acknowledged that 1in73 million is not an impossibility, but argued that such accidents would happen "once every hundred years" and that, in a country of 15 million 2child families, it is vastly more likely that the doubledeaths 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[14]) making some families more susceptible to SIDS and the error an outcome of the ecological fallacy.[15] The likelihood of two SIDS deaths in the same family cannot be soundly estimated by squaring the likelihood of a single such death in all otherwise similar families.[16]
1in73 million 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 1in73 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's 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.
Mrs. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistakes.[17]
In 2002, Ray Hill (Mathematics professor at 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.[18]
After it was found that the forensic pathologist who had examined both babies had withheld exculpatory evidence, a higher court later quashed Sally Clark's conviction, on 29 January 2003.[19]
Sally Clark was a practising solicitor before the conviction. After her threeyear imprisonment she developed a number of serious psychiatric problems including serious alcohol dependency and died in 2007 from acute alcohol poisoning.[20][21]
See also
 Base rate fallacy – Statistical formal fallacy
 Confusion of the inverse
 Data dredging
 Lucia de Berk
 Ethics in mathematics
 False positive
 False positive paradox
 Howland will forgery trial
 Jurimetrics
 Likelihood function – Function related to statistics and probability theory
 People v. Collins
 Representativeness heuristic
 Simpson's paradox – A phenomenon in probability and statistics, in which a trend appears in groups of data but disappears when these groups are combined
References
 Fenton, Norman; Neil, Martin; Berger, Daniel (June 2016). "Bayes and the Law". Annual Review of Statistics and Its Application. 3: 51–77. Bibcode:2016AnRSA...3...51F. doi:10.1146/annurevstatistics041715033428. PMC 4934658. PMID 27398389.
 Thompson, W.C.; Shumann, E.L. (1987). "Interpretation of Statistical Evidence in Criminal Trials: The Prosecutor's Fallacy and the Defense Attorney's Fallacy". Law and Human Behavior. 2 (3): 167. doi:10.1007/BF01044641. JSTOR 1393631.
 Fountain, John; Gunby, Philip (February 2010). "Ambiguity, the Certainty Illusion, and Gigerenzer's Natural Frequency Approach to Reasoning with Inverse Probabilities" (PDF). University of Canterbury. p. 6.
 Goldacre, Ben (20061028). "Prosecuting and defending by numbers". The Guardian. Retrieved 20100522.
rarity is irrelevant, because double murder is rare too. An entire court process failed to spot the nuance of how the figure should be used. Twice.

Meester, R.; Collins, M.; Gill, R.; van Lambalgen, M. (20070505). "On the (ab)use of statistics in the legal case against the nurse Lucia de B". Law, Probability & Risk. 5 (3–4): 233–250. arXiv:math/0607340. doi:10.1093/lpr/mgm003.
[page 11] Writing E for the observed event, and H for the hypothesis of chance, Elffers calculated P(E  H) < 0.0342%, while the court seems to have concluded that P(H  E) < 0.0342%
 Rossmo, D.K. (October 2009). "Failures in Criminal Investigation: Errors of Thinking". The Police Chief. LXXVI (10). Retrieved 20100521.
The prosecutor's fallacy is more insidious because it typically happens by mistake.
 "DNA Identification in the Criminal Justice System" (PDF). Australian Institute of Criminology. 20020501. Retrieved 20100521.
 N. Scurich (2010). "Interpretative Arguments of Forensic Match Evidence: An Evidentiary Analysis". The Dartmouth Law Journal. 8 (2): 31–47. SSRN 1539107.
The idea is that each piece of evidence need not conclusively establish a proposition, but that all the evidence can be used as a mosaic to establish the proposition
 Robertson, B., & Vignaux, G. A. (1995). Interpreting evidence: Evaluating forensic evidence in the courtroom. Chichester: John Wiley and Sons.
 Rossmo, D. Kim (2009). Criminal Investigative Failures. CRC Press Taylor & Francis Group.
 Gigerenzer, G., Reckoning with Risk: Learning to Live with Uncertainty, Penguin, (2003)
 http://reporter.leeds.ac.uk/428/mead.htm
 The populationwide probability of a SIDS fatality was about 1 in 1,303; Meadow generated his 1in73 million estimate from the lesser probability of SIDS death in the Clark household, which had lower risk factors (e.g. nonsmoking). In this subpopulation he estimated the probability of a single death at 1 in 8,500. See: Joyce, H. (September 2002). "Beyond reasonable doubt" (pdf). plus.maths.org. Retrieved 20100612.. 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 biased, 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: Hill, Ray (2002). "Cot Death or Murder? – Weighing the Probabilities".
it is patently unfair to use the characteristics which basically make her a good, cleanliving, 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.
 Gene find casts doubt on double 'cot death' murders. The Observer; July 15, 2001
 Vincent Scheurer. "Convicted on Statistics?". Retrieved 20100521.
 Hill, R. (2004). "Multiple sudden infant deaths – coincidence or beyond coincidence?" (PDF). Paediatric and Perinatal Epidemiology. 18 (5): 321. doi:10.1111/j.13653016.2004.00560.x.
 "Royal Statistical Society concerned by issues raised in Sally Clark case" (PDF). 23 October 2001. Archived from the original (PDF) on 24 August 2011.
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
 The uncertainty in this range is mainly driven by uncertainty in the likelihood of killing a second child, having killed a first, see: Hill, R. (2004). "Multiple sudden infant deaths – coincidence or beyond coincidence?" (PDF). Paediatric and Perinatal Epidemiology. 18 (5): 322–323. doi:10.1111/j.13653016.2004.00560.x.
 http://www.bailii.org/ew/cases/EWCA/Crim/2003/1020.html
 Shaikh, Thair (March 17, 2007). "Sally Clark, mother wrongly convicted of killing her sons, found dead at home". The Guardian. London. Retrieved 20080925.
 http://news.bbc.co.uk/1/hi/england/essex/7082411.stm