1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

probability can be coupled with an absolutely wrong approach toreality 4 .3.2. Poisson’s jurors. Laplace, and then Poisson investigated theissue of the probabilities of mistaken legal verdicts. A certain juror cannaturally make a mistake. Laplace assigned jurors a very modestability of correct judgement: he thought that for each separatelyconsidered juror the probability of a mistake was a random variableuniformly distributed on segment [0, 1/2]. Poisson did not agree; herather believed that the probability of a correct judgement should beestimated by issuing from statistical data. The impossibility ofprecisely establishing whether rightly or not a given accused personwas found guilty presents here the greatest difficulty of a directstatistical estimate.Poisson’s ideas widely applied now also consisted in that in such asituation it was necessary to construct a statistical model with theunknown probability entering it as a parameter and to attempt todetermine it by pertinent data.Let us consider the administration of justice in more detail. The trialis based on the inquest. Denote the event consisting in that theevidence collected at the inquest was sufficient for the trial to declarethe defendant guilty by A, and the contrary event by A . Given A, allthe jurors, provided their judgement is faultless, ought to unanimouslyvote for the prosecution; otherwise (event A ) for the defence.Actually, rather often the votes are divided owing to mistakes madeby the jurors. Poisson’s main proposition was that such divisionconformed to the Bernoulli pattern. If n is the number of jurors, p, theprobability of a correct judgement of each juror, the number of votesfor the prosecution, µ, it is described in the following way.1) Given A, µ is the number of successes for the n pertinentBernoulli trials with probability of success p.2) Given A , µ is the number of failures for the same pattern.According to the French legislation, n = 12 and the defendant wasdeclared guilty if µ ≥ 7. The probability of that outcome isP g = P(A)P{µ ≥ 7/A} + P( A )P{µ ≥ 7/ A } =12 12m m 12 − m m 12−m m12− + −12−m= 7 m=7∑ ∑ (3.2)P( A) C p (1 p) [1 P( A)] C p (1 p) .Criminal statistics provides the frequency of such verdicts which isapproximately equal to P g and Poisson thoroughly checked its stabilityover the years. However, expression (3.2) includes two unknownparameters, P(A) and p. Knowing only P g , it is impossible to determinethem and it is therefore necessary to turn to statistics which willindicate not only whether defendants were found guilty or exonerated,but [in one case, see below] by how many votes as well. Thus, beingaccused exactly by seven votes has probabilityP g {µ = 7} = P(A)P{µ = 7/A} + P( A )P{µ = 7/ A } =7 7 5 7 5 7P( A) C p (1 − p) + [1 − P( A)] C p (1 − p) .(3.3)12 1227