Abstract
Our full of uncertainty world requires policy analysis to be sensitive to probability theory. This chapter discusses some basic concepts and paradoxes of probability theory and expands to a discussion of an analysis of weak DNA evidence and the error rate of the legal system.
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Notes
- 1.
People v. Puckett, No. SCN 201396 (Cal. Super. Ct. Feb. 4, 2008). The analysis here parallels that in N. Georgakopoulos, Visualizing DNA Proof, 3 Crim. L. Prac. 24 (2015–2016).
- 2.
Whether Puckett’s identification by the database corresponds to exactly one positive identification or to one or more is an important and potentially contested matter. If the entire database was searched and prosecution would only follow a single positive identification, then proceeding according to the analysis of the text, that the identification corresponds to a single positive, is correct. If the search for a match stopped upon reaching the first positive identification, or if the appearance of two or more positives would then lead to further investigation of alibis to bring them down to one who would be prosecuted, then the proper approach is to consider that the identification corresponds to one or more positives. With the actual parameters of this setting, using the analysis that Puckett’s identification corresponds to one or more positives would produce a trivially different result. In different factual settings, however, especially if the database had a much greater size, then the latter method could produce a greater probability of a false positive and lower probability of true guilt for Puckett.
- 3.
To test this equivalence in Mathematica, we can use the triple equal sign (which is the abbreviation of the equality test) on the appropriate part of the output of the probability distribution function of the binomial distribution. For a number of trials n (the size of the intersection of the database and the population) and DNA test accuracy in rejecting false identifications r, the code is:
        bd1=PDF[BinomialDistribution[n,r],n-1]         bd2=PDF[BinomialDistribution[n,1-r],1]         bd1[[1,1,1]]===bd2[[1,1,1]]
which gives \(n(1-r)r^{n-1}\) for both cases and reports that the comparison is true. The intuition behind this formula is that obtaining an outcome with one failure comes from \(n - 1\) successes, which have probability \(r^{n-1}\), and one failure, which has probability \(r - 1\). Multiplying those is not enough, however, because many ways exist to obtain only one positive. The first subject may produce it, the second, the third, or the last. Therefore, we multiply the product by that number, n, and obtain the above formula, \(n(1-r)r^{n-1}\).
- 4.
Obtain the implied accuracy of the test by solving numerically the binomial distribution as seen in the previous footnote with
FindRoot[(bd1[[1,1,1]]/.n->100)==.01,{r,.95}]
which answers .93696.
- 5.
See generally Ian Ayres & Barry Nalebuff, The Rule of Probabilities: A Practical Approach for Applying Bayes’ Rule to the Analysis of DNA Evidence, 67 Stan. L. Rev. 1447 (2015).
- 6.
To whit,
PDF[BinomialDistribution[n−1,r],n−2]
which produces \(\left( {n - 1} \right)\left( {1 - r} \right)r^{n - 2} .\) Again, the intuition is that the \(n - 2\) successful rejections have probability r raised to that power, which is the last term of the formula. The probability of a false positive is \(1 - r\), and the product of these two must be multiplied by the number of ways to obtain one positive which is equal to the number of trials, \(n - 1\).
- 7.
The National Research Council explains that it cannot propose such a probability of error:
There has been much publicity about … errors made by Cellmark in 1988 and 1989, the first years of its operation. Two matching errors were made in comparing 125 test samples, for an error rate of 1.6% in that batch. The causes of the two errors were discovered, and sample-handling procedures were modified to prevent their recurrence. There have been no errors in 450 additional tests through 1994. Clearly, an estimate of 0.35% (2/575) is inappropriate[ly high] as a measure of the chance of error at Cellmark today.
The National Research Council, The Evaluation of Forensic DNA Evidence 86 (1996).
Rather, the implied error rate should be much smaller, especially assuming the recommended safeguards that include repeat testing by different laboratories.
- 8.
Id.
- 9.
See, e.g., David N. Dorfman, Proving the Lie: Litigating Police Credibility, 26 Am. J. Crim. L. 455 (1999); Christopher Slobogin, Testilying: Police Perjury and What to Do About It, 67 U. Colo. L. Rev. 1037, n. 14 (1996); Myron W. Orfield, Jr., The Exclusionary Rule and Deterrence: An Empirical Study of Chicago Narcotics Officers, 54 U. Chi. L. Rev. 1016 (1987). Alan Dershowitz points out:
I have seen trial judges pretend to believe officers whose testimony is contradicted by common sense, documentary evidence and even unambiguous tape recordings … Some judges refuse to close their eyes to perjury, but they are the rare exception to the rule of blindness, deafness and muteness that guides the vast majority of judges and prosecutors.
Alan N. Dershowitz, Controlling the Cops; Accomplices to Perjury, N.Y. Times, May 2, 1994, at A17.
- 10.
Estimates from 2004 place the proportion of plea bargains at over 97% of federal convictions and a similar estimate for state convictions. See Jed S. Rakoff, Why Innocent People Plead Guilty, New York Review of Books (Nov. 20, 2014), http://www.nybooks.com/articles/2014/11/20/why-innocent-people-plead-guilty/ [perma.cc/ZBU3-JAFF.com.F].
- 11.
D. Michael Risinger, Innocents Convicted: An Empirically Justified Factual Wrongful Conviction Rate, 97 J. Crim. L. & Crim. 761–806 (2007).
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Georgakopoulos, N.L. (2018). Probability Theory: Imperfect Observations. In: Illustrating Finance Policy with Mathematica. Quantitative Perspectives on Behavioral Economics and Finance. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-95372-4_8
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