# II. that defendant is guilty. An especially

II. Prosecutor’s’ Fallacy1This is defined as the assumption that prior probability ofrandom match is equal to probability that defendant is guilty.An especially commonexample is to confuse:the probability of an evidence (E) giventhe hypothesis (H) with the probability of a hypothesis(H) given the evidence (E). In other words P(E|H) is confused with P(H|E).

Example: Supposethat the evidence (E) is that blood type matching the defendant’s is found atthe scene of the crime. This blood type is found in approximately 1 in every1000 people.Then thestatement: the probability of thisevidence given the defendant is not the source is 1 in 1000 (i.e. P(E|H)=1/1000 where we are assuming His the statement ‘defendant is not the source’) is reasonable.

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However,it becomes the Prosecutor’s Fallacy when the prosecutors conclude that: the probability the defendant is not thesource given this evidence is 1 in 1000 (P(H|E)=1/1000),as it suggests that there is an equally small probability that the defendant isnot the source as there is the probability of observing the match in a randomperson. For thesake of simplicity, let’s limit the population size of potential blood sourceto only 10,000 people, though in real life, actual population size should benoted. Failure to take this into account could lead to Base Rate Fallacy as genericinformation is ignored.2Figure 2:Potential source population (10,000 people) As shown in Figure 2, therewill only be one actual blood source but because of the 1 in 1000 blood matchprobability, about 10 out of the other 9,999 people will have the matchingblood type. This means there is a probability of 10/11 (about 91% chance) thata person with the matching blood type is notthe source. In other words P(H|E) is 0.91 and not 1 in 1000, as claimed by the prosecution.

Bayes’ theorem can also beused to formally arrive at the same conclusion. It calculates P(H|E) in terms of P(E|H).             P(H|E) =                           =   From the above example wecan obtain the following:1)   Prior P(H) = 2)   P(E|H)=3)   P(E|not H)=1,assuming that the blood will certainly match if the defendant is the source.4)   Since P(H)=9,999/10,000 then.

. 5)   P(not H)=1/10,000 Substituting these valuesinto Bayes Theorem:            P(H|E)=              =                        =0.91 From this example, we seethat it is easy to fall into the trap of the Prosecutor’s Fallacy. It is commonto believe that since the probability of matching an innocent person’s DNA isvery low, if a person’s DNA is found to be matched the crime scene DNA, thenthe person is almost certainly the criminal.The size of the DNAdatabase is certainly significant as when it increases in size, more randommatches of DNA should be expected.10        E. Flaws of Bayes’ Theorem in Law and CourtroomsThe useof Bayes’ Theorem in legal cases have long been controversial for several ofthese reasons: 1)    Assigning a subjective priorprobability to the ultimate hypothesis is unreliable. This is because priorprobabilities are often determined from past information or past events thatmay have slightly different parameters e.

g. the probability of being guilty of murder given a history of domestic abuseand the probability of being guilty ofmurder given no history of domestic abuse have different parameters andthus cannot be seen as the same event. Prior probabilities could also be purelydetermined by an expert’s subjective assessment, which are subjected to flaws.  2)    Not all evidence can be considered orvalued in probabilistic terms. An example would be the type of voice thesuspect has (raspy, high-pitched, deep) because there is no database toaccurately form a probability for it.

3)    Due to the complexity of cases andnon-sequential nature of evidence presentation, any application of Bayes wouldbe too cumbersome for a jury to use effectively and efficiently. “Indeed,we believe that for most people – and this includes from our own experiencehighly intelligent barristers, judges and surgeons, any attempt to use Bayestheorem to explain a fallacy is completely hopeless.”3 F.Likelihood Ratio The Likelihood Ratio is a branch to Bayes’ Theorem to avoidfallacies such as Prosecutor’s Fallacy. It is commonly used as it tackles theproblem of assigning prior probabilities (referto 1 in part E). Notations, using a simple example of DNA at a crime scene. H= Hypothesis (DNA found at crime scene belongs tothe defendant)False P(H’): 0.999True P(H): 0.

001E= Evidence (DNA of Defendant is found at the crimescene)  –      False, given that the source of DNA found atcrime scene does not belong to the defendant -P(E’|H’): 0.999-      False, given that the source of DNA found atcrime scene does belong to the defendant – P(E’|H): 0.001-      True, given that the source of DNA found atcrime scene does not belong to the defendant – P(E|H’): 0.000-      True, given that given that the source of DNAfound at crime scene does belong to the defendant – P(E|H): 1.000 The Prosecutor’sFallacy in this case would be assuming that P(H’|E) and P(E|H’) arethe same. A prosecutor may make the claim that the probability the DNA did notbelong to the defendant is 1 in 1000 (0.

001), when in reality, it is 91% (0.91).TheDefendant’s Fallacy would be if the defense argues that the evidence shouldbe ignored since P(H’|E) is still low after taking into accountthe prior and the evidence.  All thatbeing said, the Likelihood Ratio is the Prosecution Likelihood forEvidence divided by the Defendant Likelihood for Evidence:  The ProsecutionLikelihood for Evidence would be seeing the evidence E if H is true (defendant is guilty). In notationform, P(E|H), which is 1. The DefendantLikelihood for Evidence would be the probability of seeingthe evidence if H is not true (defendant isnot guilty). In notation form, P(E|H’), which is 0.

001.  From this, the Likelihood ratio can be calculated: = 1,000This means that the prior odds of H are not as significantbecause regardless, the posterior odds of H mustincrease by a factor of 1,000 as a result of seeing the evidence. In general, if the likelihoodratio is more than 1, E results in ahigher posterior probability of H. If E is less than 1, it results in a lower posterior probability of H.If the likelihood ratio = 1, E isnot valuable as the posterior probability calculated will be the same,according to the above formula.  G. Alternative toBayes’ Theorem in law and courtroomsHypothesis TestingThis is used to make decisions in the courtroom, morespecifically to determine whether the null hypothesis of being innocent can berejected at an appropriate level of significance.

Steps1. State the hypothesis. In courtrooms, thenull hypothesis (which is that the defendant is innocent) and the alternative hypothesis (this must bethe opposite of null hypothesis, which is that the defendant is guilty) must beestablished. The null hypothesis is assumed until proved otherwise, beyondreasonable doubt.   2. Set levelof significance.

Inhypothesis testing, data is collected to prove the null hypothesis wrong(defendant is innocent) by basing it on the likelihood of selecting a samplemean from a population. Inbehavioral science, the criterion or level of significance is typically set at5%. If the probability of obtaining a sample mean is less than 5% then the valuestated in the null hypothesis is rejected.1 Thompson,E.L.; Shumann, E. L.

(1987). “Interpretationof Statistical Evidence in Criminal Trials: The Prosecutor’s Fallacy and theDefense Attorney’s Fallacy”. Law and Human Behavior2 Fenton, N., Neil,M., Berger,D., (March 9, 2016) Bayes and the Law.3 N. Fenton& M.

Neil (2011)

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