Jurors may be a priori biased for against the prosecution in...

Jurors may be a priori biased for against the prosecution in a criminal trial. Each juror is questioned by both the prosecution and the defense (the voir dire process), but this may not reveal bias. Even if bias is revealed, the judge may not excuse the juror for cause because of the narrow legal definition of bias. For a randomly selected candidate for the jury, define events $B_{0}$ , $B_{1},$ and $B_{2}$ as the juror being unbiased, biased against the prosecution, and biased against the defense, respectively. Also let $C$ be the event that bias is revealed during the questioning and $D$ be the event that the juror is eliminated for cause. Let $b_{i}=P\left(B_{i}\right)$ for $i=0,1,2$ $c=P(C | B_{1})=P(C | B_{2}),$ and $d=P(D | B_{1} \cap C)=P(D | B_{2} \cap C)$ ["Fair Number of Peremptory Challenges in Jury Trials," $J .$ Amer. Statist. Assoc. $, 1979 : 747-7533 ]$ (a) If a juror survives the voir dire process, what is the probability that he/she is unbiased (in terms of the $b_{i} s, c,$ and $d ) ?$ What is the probability that he/she is biased against the prosecution? What is the probability that he/she is biased against the defense? [Hint: Represent this situation using a tree diagram with three generations of branches.] (b) What are the probabilities requested in (a) if $b_{0}=.50, b_{1}=.10, b_{2}=.40$ (all based on data relating to the famous trial of the Florida murderer Ted Bundy), $c=.85$ (corresponding to the extensive questioning appropriate in a capital case), and $d=.7$ (a "moderate "judge)?

Jurors may be a priori biased for against the prosecution in a criminal trial. Each juror is questioned by both the prosecution and the defense (the voir dire process), but this may not reveal bias. Even if bias is revealed, the judge may not excuse the juror for cause because of the narrow legal definition of bias. For a randomly selected candidate for the jury, define events $B_{0}$ , $B_{1},$ and $B_{2}$ as the juror being unbiased, biased against the prosecution, and biased against the defense, respectively. Also let $C$ be the event that bias is revealed during the questioning and $D$ be the event that the juror is eliminated for cause. Let $b_{i}=P\left(B_{i}\right)$ for $i=0,1,2$ $c=P(C | B_{1})=P(C | B_{2}),$ and $d=P(D | B_{1} \cap C)=P(D | B_{2} \cap C)$ ["Fair Number of Peremptory Challenges in Jury Trials," $J .$ Amer. Statist. Assoc. $, 1979 : 747-7533 ]$
(a) If a juror survives the voir dire process, what is the probability that he/she is unbiased (in terms of the $b_{i} s, c,$ and $d ) ?$ What is the probability that he/she is biased against the prosecution? What is the probability that he/she is biased against the defense? [Hint: Represent this situation using a tree diagram with three generations of branches.]
(b) What are the probabilities requested in (a) if $b_{0}=.50, b_{1}=.10, b_{2}=.40$ (all based on data relating to the famous trial of the Florida murderer Ted Bundy), $c=.85$ (corresponding to the extensive questioning appropriate in a capital case), and $d=.7$ (a "moderate "judge)?

Key Concepts

Law of Total Probability

The law of total probability is used to calculate the probability of an event by considering all mutually exclusive ways that the event can occur. In the context of juror selection, it is applied to aggregate the probabilities over the different potential bias states (unbiased, biased against the prosecution, or biased against the defense) and their corresponding outcome probabilities (such as bias being revealed or not). This law is essential for deriving overall probabilities that serve as denominators in conditional probability calculations.

Bayes' Theorem

Bayes' theorem is a fundamental result in probability theory that provides a way to update the probability estimate for an event based on new evidence. In this problem, it is applied to compute the posterior probability that a juror is unbiased or biased, given the outcome of the voir dire process. By combining the prior probabilities of the juror's bias with the conditional probabilities of bias revelation and elimination, Bayes' theorem allows one to derive these updated probability assessments.

Tree Diagrams

Tree diagrams are visual tools used to represent all possible outcomes of a sequential process, where each branch represents a possible event and its associated probability. In this context, the tree diagram helps to break down the multi?stage process of juror selection, bias revelation, and potential elimination, thereby organizing the different conditional probabilities involved and making it easier to apply formulas like the law of total probability and Bayes' theorem.

Conditional Probability

Conditional probability is the concept of computing the likelihood of an event occurring given that another event has already occurred. In the context of this problem, it is used to determine the updated probability that a juror is unbiased or biased, based on the information that they have survived the voir dire process. This approach underpins the process of updating beliefs when additional evidence or outcomes (such as revealed bias or elimination for cause) are observed.

Transcript

00:01 In this question, we have a juror selection process where events b0, b1, and b2 represent the events that the juror being considered is either not biased, which is b0, or biased against the prosecution, which is b1, or biased against the defense, which is b2.

00:24 If they are biased against either the prosecution or the defense, the probability of the event c, which is the event that they're biased, is revealed in questioning occurs with probability c.

00:43 And given that they are either biased against the prosecution or the defense, and this bias has been revealed in questioning, the probability of event d, which is their elimination from the voir dire process, is equal to d.

01:04 And now part a of the question asks, what is the conditional probability of the jurors ' type of bias, conditional on having survived the voir dire process.

01:17 So it's looking for the probability that the juror has no bias, given that they survived the voir dire process, which means that de -compliment, which will denote d -prime, occurred.

01:35 And we're also asked for the other conditional probabilities for juror bias.

01:52 Now, these are all calculated in an identical manner.

01:55 So in this tutorial, perhaps we will just solve for one, and then i can present the answers for the other two, knowing that they are calculated in the same way.

02:12 So to solve for this conditional probability, we apply bays theorem, which says that if a set of events form a partition of the sample space, and the events describing the jurors bias do indeed form a partition of the sample space, the juror can either have no bias, can be bias against the prosecution or bias against the defense, but there is no other option, and they have to be one of these.

02:44 Given that the events be, the b events form a partition of the sample space, then for some other event, say event d prime, the probability of one of the b events conditioned on d prime is equal to the following.

03:02 And now we're asked to provide this probability in terms of these probabilities given in the question.

04:07 So we already know some of the values.

04:11 So we know the probability of b2 is given by this value here.

04:15 And we already have this one and this one and this one.

04:18 So we must find the conditional probabilities that we see in the numerator and denominator.

04:25 So one thing that is suggested in the question is to use an event tree to help us see our.

04:32 Solution.

04:34 So let's do that.

04:35 Suppose we have a juror.

04:38 Now in terms of bias, there are only three options.

04:44 So the juror may have no bias, may be biased against the prosecution, or may be biased against the defense.

04:58 And the probabilities for each of these are b0, b1, and b2.

05:06 Now, if the juror has no bias, they can only not have their bias revealed in questioning.

05:15 So that's c prime.

05:16 With probability one.

05:19 So they have no bias so it cannot be revealed.

05:25 If they're biased against the prosecution, they may either have their bias revealed during the voir dire process or not revealed.

05:40 And from the question these probabilities are c and so this probability would be one minus c.

05:49 And the same thing applies to a juror who has a bias against the defense.

06:00 Now if a juror is both not biased and therefore does not have their bias revealed in the questioning, they will survive the voir dire process...

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