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1. Consider the following simplified view of a jury weighing evidence (treat the jury as a single decision maker). A defendant in a court case is accused of committing a crime. Independent of guilt (G) or innocence (I), with probability 3/4, the defendant possesses evidence. This evidence is in the set {d₁, d₂, d₂}. When he possesses evidence, the probability of which document it is depends on whether he is guilty or innocent. We denote the probabilities with which d, is realized, conditional on I or G, respectively, (and conditional on evidence being realized) by P₁, and PG₁. Suppose these are given by P₁3 = 3/8, P12 = 3/8, P₁1 = 1/4, PG3 = 1/8, PG2 = 1/8, and PGI = 3/4. Assume the prior belief that the defendant is innocent is 3/8. Suppose that the jury believes that both guilty and innocent types of the defendant do not disclose d,, but both do disclose d₂ and d₂. What is the jury's posterior belief that the defendant is innocent when it observes no document disclosed?

1. Consider the following simplified view of a jury weighing evidence (treat the jury as a single decision maker). A defendant in a court case is accused of committing a crime. Independent of guilt (G) or innocence (I), with probability 3/4, the defendant possesses evidence. This evidence is in the set {d₁, d₂, d₂}. When he possesses evidence, the probability of which document it is depends on whether he is guilty or innocent. We denote the probabilities with which d, is realized, conditional on I or G, respectively, (and conditional on evidence being realized) by P₁, and PG₁. Suppose these are given by P₁3 = 3/8, P12 = 3/8, P₁1 = 1/4, PG3 = 1/8, PG2 = 1/8, and PGI = 3/4. Assume the prior belief that the defendant is innocent is 3/8. Suppose that the jury believes that both guilty and innocent types of the defendant do not disclose d,, but both do disclose d₂ and d₂. What is the jury's posterior belief that the defendant is innocent when it observes no document disclosed?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Bayesian Probability in Legal Contexts: A Simplified Jury Decision-Making Model #### Problem Statement: Consider a simplified model where a jury must weigh evidence to decide on the guilt or innocence of a defendant in a court case, treating the jury as a single decision-maker. The defendant is accused of committing a crime. The probabilities are as follows: - The defendant can either be guilty (G) or innocent (I). - The probability that the defendant is guilty is \(\frac{3}{4}\) and innocent is \(\frac{1}{4}\). When evidence is possessed, it can be one of the documents in the set \(\{d_1, d_2, d_3\}\). The occurrence of these documents is dependent on whether the defendant is guilty or innocent. The probabilities are conditional on guilt or innocence and are denoted by \(p_{Ii}\) and \(p_{Gi}\), respectively. Given probabilities: - \(p_{I3} = \frac{3}{8}\) - \(p_{I2} = \frac{3}{8}\) - \(p_{I1} = \frac{1}{4}\) - \(p_{G3} = \frac{1}{8}\) - \(p_{G2} = \frac{1}{8}\) - \(p_{G1} = \frac{3}{4}\) #### Problem Solving: Assume that the prior belief (prior probability) that the defendant is innocent is \(\frac{3}{8}\). Suppose the jury believes that both guilty and innocent defendants do not disclose \(d_1\), but both disclose \(d_2\) and \(d_3\). The question posed is: What is the jury’s posterior belief that the defendant is innocent, given that no document is disclosed? To solve this, we must employ Bayesian updating to determine the posterior probability of innocence given no evidence document is disclosed.
Transcribed Image Text:### Bayesian Probability in Legal Contexts: A Simplified Jury Decision-Making Model #### Problem Statement: Consider a simplified model where a jury must weigh evidence to decide on the guilt or innocence of a defendant in a court case, treating the jury as a single decision-maker. The defendant is accused of committing a crime. The probabilities are as follows: - The defendant can either be guilty (G) or innocent (I). - The probability that the defendant is guilty is \(\frac{3}{4}\) and innocent is \(\frac{1}{4}\). When evidence is possessed, it can be one of the documents in the set \(\{d_1, d_2, d_3\}\). The occurrence of these documents is dependent on whether the defendant is guilty or innocent. The probabilities are conditional on guilt or innocence and are denoted by \(p_{Ii}\) and \(p_{Gi}\), respectively. Given probabilities: - \(p_{I3} = \frac{3}{8}\) - \(p_{I2} = \frac{3}{8}\) - \(p_{I1} = \frac{1}{4}\) - \(p_{G3} = \frac{1}{8}\) - \(p_{G2} = \frac{1}{8}\) - \(p_{G1} = \frac{3}{4}\) #### Problem Solving: Assume that the prior belief (prior probability) that the defendant is innocent is \(\frac{3}{8}\). Suppose the jury believes that both guilty and innocent defendants do not disclose \(d_1\), but both disclose \(d_2\) and \(d_3\). The question posed is: What is the jury’s posterior belief that the defendant is innocent, given that no document is disclosed? To solve this, we must employ Bayesian updating to determine the posterior probability of innocence given no evidence document is disclosed.
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