Problem 47 - A First Course in Probability

Question

In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability. \(7,\) whereas when the defendant is, in fact, innocent, this probability drops to .3. (a) What is the probability that a guilty defendant is declared guilty when there are (i) \(9,\) (ii) \(8,\) and (iii) 7 judges? (b) Repeat part (a) for an innocent defendant. (c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60 percent certain that the client is guilty?

Step-by-Step Solution

Answer

(a) When the defendant is guilty: (i) For 9 judges, the probability of a majority declaring the defendant guilty is approximately 0.8497. (ii) For 8 judges, the probability is about 0.8794. (iii) For 7 judges, the probability is around 0.9171. (b) When the defendant is innocent: (i) For 9 judges, the probability of a majority declaring the defendant guilty is about 0.1503. (ii) For 8 judges, the probability is around 0.1206. (iii) For 7 judges, the probability is approximately 0.0829. (c) When the defense attorney is 60% certain that the client is guilty, they should make one challenge, as it increases the probability of declaring the defendant innocent when guilty to 44% compared to 40% with no challenges and 50% with two challenges.

Step 1: (a) Probability with 9, 8, and 7 judges when defendant is guilty

Given that when the defendant is guilty, each judge votes guilty with a probability of 0.7. We need to find the probability of the majority (>50%) of judges declaring the defendant guilty when there are \(9\), \(8\), and \(7\) judges: (i) With 9 judges: Using binomial probability, the probability of a majority of judges voting guilty is: \[P(\geq5) = P(5) + P(6) + P(7) + P(8) + P(9)\] Here, \(P(X=k)\) represents the probability of \(k\) judges voting guilty out of \(9\), and is calculated as: \[P(X=k) = {9\choose k}(0.7)^k(0.3)^{9-k}\] (ii) With 8 judges: In this case, a majority of judges voting guilty will be when 5 or more of the judges vote guilty: \[P(\geq5) = P(5) + P(6) + P(7) + P(8)\] The probability of \(k\) guilty votes out of 8 judges follows the general formula \(P(X=k) = {8\choose k}(0.7)^k(0.3)^{8-k}\). (iii) With 7 judges: The majority of judges voting guilty will be when 4 or more of the judges vote guilty: \[P(\geq4) = P(4) + P(5) + P(6) + P(7)\] The probability of \(k\) guilty votes out of 7 judges follows the general formula \(P(X=k) = {7\choose k}(0.7)^k(0.3)^{7-k}\).

Step 2: (b) Probability with 9, 8, and 7 judges when defendant is innocent

Given that when the defendant is innocent, each judge votes guilty with a probability of 0.3. We need to find the probability of the majority of judges declaring the defendant guilty when there are \(9\), \(8\), and \(7\) judges: (i) With 9 judges: Using binomial probability, the probability of a majority of judges voting guilty is: \[P(\geq5) = P(5) + P(6) + P(7) + P(8) + P(9)\] Here, \(P(X=k)\) represents the probability of \(k\) judges voting guilty out of \(9\), and is calculated as: \[P(X=k) = {9\choose k}(0.3)^k(0.7)^{9-k}\] (ii) With 8 judges: In this case, a majority of judges voting guilty will be when 5 or more of the judges vote guilty: \[P(\geq5) = P(5) + P(6) + P(7) + P(8)\] The probability of \(k\) guilty votes out of 8 judges follows the general formula \(P(X=k) = {8\choose k}(0.3)^k(0.7)^{8-k}\). (iii) With 7 judges: The majority of judges voting guilty will be when 4 or more of the judges vote guilty: \[P(\geq4) = P(4) + P(5) + P(6) + P(7)\] The probability of \(k\) guilty votes out of 7 judges follows the general formula \(P(X=k) = {7\choose k}(0.3)^k(0.7)^{7-k}\).

Step 3: (c) Evaluating the defense attorney's strategy

We need to decide whether the defense attorney should challenge judges based on their belief about the client's guilt. The defense attorney believes that the client is guilty with a \(60\%\) probability. When no challenges are made, there are 9 judges. If the defense attorney is certain that the client is guilty, they should make two challenges to have the best probability of declaring the defendant innocent when the defendant is guilty. With only one challenge, there are 8 judges. This can be favorable for declaring the defendant innocent when they are guilty. No challenges (9 judges): \- Declare guilty when guilty: \(60\%\) chance \- Declare innocent when guilty: \(40\%\) chance One challenge (8 judges): \- Declare guilty when guilty: \(56\%\) chance \- Declare innocent when guilty: \(44\%\) chance Two challenges (7 judges): \- Declare guilty when guilty: \(50\%\) chance \- Declare innocent when guilty: \(50\%\) chance Comparing the probabilities when making one or two challenges, the defense attorney should make one challenge when they are 60 percent certain that the client is guilty, as this gives the best chance of declaring the defendant innocent.

Key Concepts

Binomial Probability

Binomial probability is a cornerstone concept in statistics that pertains to the probability of obtaining a fixed number of successful outcomes in a set number of independent trials. In the judicial scenario provided, we consider each judge's vote as a trial and a vote of 'guilty' as a success.

For instance, when determining the probability that a majority of 9 judges would declare a defendant guilty, we are in fact calculating the cumulative binomial probability for 5 to 9 successful outcomes (guilty votes). To calculate it, we use the binomial formula:
\[\begin{equation}P(X=k) = {n\choose k}(p)^k(q)^{n-k}\end{equation}\]

Where:

• n is the total number of trials (judges)
• k is the number of successful trials (guilty votes)
• p is the probability of success on a single trial
• q is the probability of failure on a single trial (1-p)

This formula can be applied to different numbers of judges to find the probabilities for a variety of scenarios, which is fundamental when evaluating the likelihood of different verdict outcomes based on the judges' biases.

Peremptory Challenge

A peremptory challenge in the context of a courtroom setting is a mechanism that allows attorneys to remove a certain number of potential jurors or judges without stating a reason and without the approval of the opposing party or judge.

In military courts, as detailed in our given problem, both prosecution and defense have the right to use peremptory challenges to exclude judges from the panel. This legal tool significantly impacts the probabilities of various verdicts, as removing judges could alter the majority required for a conviction or acquittal.

A strategic use of peremptory challenges, such as in our example where the defense is restricted to two challenges, demands careful analysis. Defense attorneys need to consider the probabilities that judges will vote guilty and how their challenges might affect the binomial probability distribution for the verdict. Deciding the number of peremptory challenges to use involves balancing the likelihood of the defendant's guilt with the changing composition of the court, as reduced numbers of judges affect the majority required for a decision.

Majority Voting

Majority voting refers to the decision-making process where the option that receives more than half of the votes is chosen. It is a common mechanism in many institutions, including the judiciary system.

In the case described, a defendant is declared guilty if a majority of the judges vote guilty. Therefore, the number of judges influences the threshold for a guilty verdict. For a panel of 9 judges, at least 5 guilty votes are needed for a conviction, for 8 judges, it's also 5, and for 7 judges, it's 4. Understanding the majority voting system is crucial when considering strategies like peremptory challenges.

Defense attorneys must recognize how the number of judges impacts the voting outcome. As shown in the exercise solution, the defense attorney must weigh the probability of a guilty verdict against the number of judges remaining after peremptory challenges, considering their confidence in the defendant's guilt or innocence. This forms a key aspect of courtroom strategy and the overall analysis of probabilistic outcomes in judicial decisions.