An assesment on the accuracy behind a jury’s veredict in a trial

I’ve spent my entire childhood watching TV shows about lawyers, the White House, politicians, the FBI, and the CIA. Following the development of cases, the implication of criminals, and then deciphering whether someone was innocent or not, really caught my attention.

 

The jury radiated this intriguing and mysterious aura that captured me. They were strangers with unknown names and lives, that were about to make a life-or-death decision for someone. The fact that a mystifying group of people had such a predominant role in the outcome of a case, was mind-blowing. Nonetheless, it was not until some years later when I watched a series that focused on the study of a jury, that I started to appreciate it more. The action revolved around creating a perfect replica of the jury to practice the trial, therefore predicting the effect the attorney's speech could have on the jury. Being obsessed with human behavior and seeing the psychological and human approach that the plot irradiated, left me forever intrigue with this figure. And now, some years later, when the curiosity hasn’t left me yet, I use the jury as the base of my investigation.

 

The jury pool is a group of people created by randomly choosing voters and licensed driver citizens, to ensure that they are overage and participate in the life of the country (Reinhart, 1998). From that list, only some of them will be finally summoned for jury service, to take part in the decision-making of a case. The final candidates must be selected bearing in mind their ability to judge impartially. To do so, they go through a process called voir dire. This procedure aims to 4 exclude anyone who cannot partially judge because they know a member involved in the case, have information about the case, or have some kind of prejudice towards one of the parties. Those will be immediately dismissed by the judge (United States Courts, 2023). In this investigation, the jury model employed is based on the United States judicial system, recognizing the differences with other countries. The selection of this country is deliberate, as its jury system is highly regarded, making the conclusions of this investigation more relevant to its legal context.

 

At the end of the day, the jury’s role is to dictate someone’s future based on the evidence presented. The decision between guilty or innocent will be based on the vast amount of evidence and testimony presented at trial, but that decision is ultimately a matter of luck and intuition. The fact that humans constitute the jury, guarantees that the decision might come with a percentage of error. There are limitations in perception and reasoning which can be influenced by factors such as emotions, biases, or lack of information. This is why the probability that the jury is 100% right all the time, seems impossible and far away from reality. Mathematics, on the other hand, can be an accurate and precise tool to model that reality. In this case, by calculating the probability of a verdict being the right one, we are not only learning about the strengths or weaknesses of the judicial system but to what extent we can trust a jury in a trial. In general terms, mathematics can help us to understand and improve the world around us, being the base of everything that is about to come.

 

This investigation aims to determine, relying on a mathematical approach, whether the jury's verdict should be trusted to be the correct one. To answer this question, we will model the jury’s 5 performance with Condorcet’s Jury Theorem, and then proportionate the possible effects that probability can have on a real jury.

The Condorcet’s theorem was first expressed by the French philosopher Marquis de Condorcet in 1785 in his work Essay on the Application of Analysis to the Probability of Majority Decisions (Condorcet, 1785). In his essay, he presented a mathematical respond to the likelihood that a group of people reached a correct decision. In principle, the theorem was created to be applied to many areas with group decisions (Siscoe, 2022), but ended fitting in the jury world perfectly and adopted the name of Condorcet’s Jury Theorem.

 

In this investigation Condorcet’s theorem will be applied to the following hypothetical situation: at the end of a trial and after hearing all the testimonies and evidence, we assume that a decision must be made between two options, whether the defendant is guilty (G) or innocent (I). We also assume that one of the options is correct, but we do not know which one. Furthermore, there is a number of people (the individuals on the jury) defined as n who must make the decision. Each one will vote G or I, and the final decision will be G or I based on the sum of all the individual responses.

 

The theorem follows a cumulative binomial distribution, which means it summarizes the probability that a value will take one of two independent values (success or failure) under a given set of assumptions (Barone, 2023). To prove it, let’s suppose X (a binomial random variable) is the number of jurors that correctly identify if the judged person is guilty. Now suppose we have n members of the jury making the decision. The probability that they get to the right answer is the same for all the jurors, and each juror is independent of every other juror. This is an example of a binomial experiment, because as we know from statistics and probability, a binomial experiment requires that the probability of success is consistent for all trials, the trials are independent, and there are only two possible results, and these three conditions are met here. The probability distribution of X is called the binomial distribution and we can express it mathematically as X~B (n, p). The fact that some conditions need to be present for the formula to be accurate shows a limitation and explains why some researchers have worked to create variations and improvements of the model created by Condorcet. Two assumptions need to happen (Statistical Consultants Ltd, 2010) -

 

Assumption 1 - Every member of the jury must vote independently, without taking into consideration the decision of the rest. This way it is avoided that the decision is influenced, which would complicate the possible outcomes of the problem. However, in practice, it can be challenging to ensure complete independence and lack of influence among jurors. It is mostly idealistic to pretend that a human being takes a decision exclusively based on what was heard during the trial because factors such as persuasive arguments from fellow jurors, or social pressure may influence their individual opinions.

 

Assumption 2 - Every juror n has a probability \(P_n\) of correctly identifying if the judged person is guilty or innocent. All the jurors have the same probability, therefore being competent (which is already contemplated during the voir dire). A competent juror is considered to give the correct verdict at least more than half of the times they participate in a trial. Therefore, we assume that for each juror \(P_n >\frac{1}{2}\) . Nevertheless, in a real jury, it is very difficult if not impossible for everyone to have the same probability of choosing correctly. Each of the individuals has a different background and a role in society, they will tend to position themselves on the side of someone they feel represented. In a case of sexual harassment, previous victims of these same circumstances will have more probability of showing signs of sympathy for the victim and therefore vote in their favor. Therefore, we might assume an average is used to avoid tedious results, even if it's not 100% accurate.

Based on the previous assumptions, we know that each of the jurors will have more than a 50% probability of identifying if the judged person is guilty or innocent, and equal to that of their peers. Chosen randomly, we assume it is 60% for the first model of the situation. Then, we will consider other probabilities to compare and analyze the sensibility of that value to the final probability of correctness.

 

If there was only one member of the jury, the chance of arriving at the correct answer would be 60%, while if there were 3, we should consider the probability of getting the correct answer given the skills of each individual n. The different possible outcomes for such a situation with three members in a jury are shown in figure 2.

 

Where -

S - Success = the jury members declare innocence when the defendant is innocent, and guilt when the defendant is guilty.

F -  Failure = the jury members fail to declare guilt when the defendant is guilty, and innocence when the defendant is innocent.

Figure 2 represents a decision tree that shows all the possible outcomes in a jury of three members if they all give their verdict on a case. To calculate the probability of correctness for each of the trials we would multiply the individual probabilities as seen, multiply because they are independent. And to see the actual probability of choosing correctly with three jurors, we would just have to add the probabilities of only those trials in which the answer was correct. This means we are only going to consider the outcomes according to the value of X = 2 and X = 3, at least two correct jurors. In our case, this is adding [1] + [2] + [3] + [5] giving as a result -

 

P(X = 2) = P(SSF or SFS or FSS) = [2] or [3] or [5] = 3(0.6 x 0.4 x 0.6) = 3(0.144)

P(X = 3) = P(SSS) = [1] = 0.6 x 0.6 x 0.6 = 0.216

P(X = 2) + P(X = 3) = 3(0.144) + 0.216 = 0.648 = 64.8%

 

64.8% is the percentage of correctness for a jury of three members. When compared to the correctness percentage of a single juror, which is 60% in this scenario, we could assume that as the number of jurors increases, the likelihood of being correct also increases. However, this assumption will be further examined through a graph depicting correctness probabilities across various jury sizes. Overall, this limited increase in the probability of correctness compared to that of a single juror explains why juries are formed with more than three members; because, if not, just half of the cases judged, would be correctly resolved.

 

In a real jury, this situation is modelled by the Condorcet’s theorem, a generalization of a particular case, like seen in the previous example. It calculates the probability, \(P_{g_n}\), that the jury gives the correct answer, with the formula (Statistical Consultants Ltd, 2010)-

 

\(P_{g_n} = \sum_{k=m}^{n} \left[ \frac{n!}{(n-k)! \, k!} \right] \left( p^k (1-p)^{n-k} \right)\)

 

Where -

P- probability of success or probability of the jury being right, 0 ≤ P ≤ 1, P ∈ R

g -  indicates group probability

n - number of people in the jury, n  ≥ 1, n ∈ N

m - the number of jurors required for a majority. When the total number of jurors (n ) is even, the formula is  \(\frac{n}{2} + 1\) = m. When the total number of jurors (n) is odd, the formula is \(\frac{n}{2} + 1\). By using these formulas, we are ensuring that m ∈ N, which aligns with the idea of a majority being a whole number of jurors and not a fractional or decimal value. K - the index, K ∈ N

n!: the product of the first n positive integers, n! ∈ N

 

This theorem follows the distribution of a binomial random variable, where \(\frac{n!}{(n-k)!k!}\) can be abbreviated into \(\binom{n}{k}\), also known as the binomial coefficient, to express the total ways in which we can choose K voters or jurors that are going to be correct from among n possible voters. We can easily evaluate \(\binom{n}{k}\) on our graphic calculator. That is later multiplied by the probability that K voters are correct, times the probability that the remaining n – K voters are incorrect. We are then told that the sum of the binomial of n jurors starting from a majority, m, gives us the probability of being correct.

 

In general terms, the formula is composed of two concepts that multiply.\(\frac{n!}{(n-k)! \, k!}\) that represents the number of ways of obtaining K successes from n trials, multiplied by \((p)^k (1-P)^{n-k}\) which is the probability of obtaining K successes and n − K failures in a particular order.

 

A jury is normally composed of twelve members in a criminal trial in the USA (United States Courts, 2023). Therefore, we substitute n by 12, as the number of members of the jury; m by \(\frac{12}{1}\)+ 1 = 7 as the number of jurors needed to reach a majority and therefore assumed to be the number of voters that at least need to be correct; and p by 0.6 as we have assumed that all the members have the same probability of being correct and this is higher than 0.5. After substituting the values, we obtain-

 

\(P_{gn} = \sum_{k=7}^{12} \left[ \frac{12!}{(12-k)! \, k!} \right] (0.6)^k (1-0.6)^{12-k} \)

 

\([ P_{g_n} = \sum_{k=7}^{12} \binom{12}{k} (0.6)^k (1 - 0.6)^{12-k} ] \)

 

\([ P_{g_n} \approx 0.665 ]\)

 

\([ P_{g_n} \approx 066.5% ]\)

 

66.5% is the percentage of correctness of a jury with 12 members, therefore applicable to every jury trial that takes place in the USA every day. The probability, as expected, has increased as the number of jurors has increased. From 3 to 12 members the probability has grown a 2.3%, a low probability if we consider the number of people has been quadrupled. What the number suggests is that only 67 cases per 100 receive the correct verdict, leaving the jury's professionality and trust under scrutiny.

 

However, although 12 is the established number of members in a jury, calculating the probability of correctness with different values could be enriching to analyze the relationship between the number of members and the probability of correctness considering all the members have a 60% probability of judging correctly. In addition, we will be able to judge if the number of members chosen is the most appropriate or if changing it should be considered to increase the success rates.

The values 6, 9, 12, 18, and 24 were selected because they all share 3 as a common factor. Additionally, they create symmetry around 12. This choice allowed the examination of diverse scenarios involving four distinct groups of jurors, two larger and two smaller than the commonly used 12-member juries in the USA judicial system.

 

Figure 3 illustrates a progressive increase in the probability of success as the jury size escalates from 6 to 24 members. This trend suggests that larger juries are more likely to arrive at the correct verdict. However, an inconsistency is evident as the probability of the jury being right decreases from 9 to 12 jurors before rising again for 18 jurors. Surprisingly, a 12-member jury is more likely to make a correct judgment than a 6-member jury but less likely than a 9, 18, or 24- member jury. The probabilities vary, with a 6-member jury having the lowest probability of 0.544, and a 24-member jury having the highest at 0.787, assuming the individual probability of making a correct judgment is 0.6. It's crucial to note that, despite some probabilities being smaller than others, none fall below 0.5. This suggests that every jury is expected to be correct in at least one out of two cases.

 

In conclusion, a 24-member jury appears to be the most accurate decision-making group, establishing it as the preferred model. Nevertheless, the practicality of implementing such large juries is questionable due to the challenges associated with reaching a unanimous agreement within such a sizable group. The United States may have chosen a 12-member jury as an optimal size to mitigate the potential challenges linked with larger groups, like 18 members, which have a higher probability of correctness, 0.737, but may encounter difficulties in reaching a consensus.

 

To determine the most appropriate jury size, we should consider jurors with varying probabilities of making a correct judgment other than 0.6, such as 0.7 and 0.8. To analyze each value, we will create a two-way table to assess the sensitivity of the data. This table will confront different jury sizes with varying probabilities of success for each member, enabling us to identify which variables contribute significantly to differences in the outcome.

The values presented in the table were calculated using the previously mentioned formula of Condorcet’s theorem. The calculation involved adjusting both the probability value and the number of jurors to align with the specificities outlined in the table.

 

Figure 4 shows that the highest probability of accurately determining the innocence or guilt of an individual rests with juries where each member has an individual probability of 0.8, regardless of the jury size. This observation aligns logically with the notion that the final decision is a cumulative outcome of individual decisions. However, there are some exceptions, notably when individual probabilities are 0.7, and the jury size is 9, 18 or 24 members. Moreover, it becomes evident that the probability of independent success has a more pronounced impact on the results than the number of members in the jury. This can be seen in the increase from 0.737 to 0.787 when transitioning from 18 to 24 members, and the increase from 0.737 to 0.940 when the individual probability increases from 0.6 to 0.7 with 18 members, this second one constituting a bigger difference. This observation highlights a limitation in our initial calculation using Condorcet’s theorem, as it exclusively considered a situation where the individual probability was fixed at 0.6.

We conducted a thorough analysis of the data by graphing it to visually represent its linearity. To quantify the correlation between variables, we calculated the Pearson correlation coefficient, a metric that helps us understand how changes in one variable correspond to changes in another in a predictable manner. In our examination, the orange line representing a probability of 0.7 showed a strong positive and linear correlation between jury size and the juror's individual likelihood of reaching the correct verdict. Specifically, as the size of the jury increases, the probability of delivering a correct verdict also increases. This correlation stands out as notably stronger when compared to the medium positive and linear correlations observed with the other two probabilities, 0.6 and 0.8. These findings provide valuable insights into the relationship between jury size and the accuracy of verdicts, with the orange line indicating a more predictable connection.

In the USA in 2021, the total number of civil and criminal cases dropped to 419,032, where 813 were criminal cases, of which 87 involved a jury (United States Courts, 2021). We can see from the data the vast difference in numbers that leaves jury trials in the minority.

 

The number of jury trials is proportionately lower but has also declined recently (United States Courts, 2023). The reasons behind his imminent demise have been considered and studied by many doctors in the area and their ideas can be summarized in three considerations-

• Clients have the perception that jury trials are very expensive in terms of time, but also of money. Even if 80% of civil attorneys believe they are worth the cost, it doesn't take away from the fact that clients are turning away from them.
• Mediation is now considered the superior resolution process among clients as it claims to be easy, fair, and cost-effective. However, the unpredictability of the jury acts as a factor against its survival, which gives mediation a point in favor as this is the new method through which the defendant and plaintiff reach agreements and prevent the case from being decided by a total of 12 people who can choose either side of the case (Townson, 2023).

 

The percentage of correctness adds to this problem, as it shows how little accurate a jury can be. What could have been used to restore the jury's fame, by arguing it is much more accurate than just mediation or just one judge deciding, turns out to have a 66.5% of correctness. It acts as an added variable, slowly pushing jury trials out of the spectrum.

In exploring how math can be applied in real-life situations, we tested it in a courtroom setting. One significant question that comes up when we watch a jury trial is whether they can really find the truth and make the fairest decision. This question becomes even more important when we consider that, in our country, juries are only used in specific cases like homicides, threats, trespassing, bribery, etc. (Barco, 2022). Initially, it seemed like judges, with their legal education and experience, might make more accurate decisions. However, our research challenged this idea, showing that involving more people in the decision-making process led to more accurate outcomes.

 

Nevertheless, a 12-member jury only made the correct decision 66.5% of the time when each juror's chance of being correct was 0.6. This percentage increased to 90.1% when individual probabilities rose to 0.8, although this seems more like an ideal scenario than something that happens in real life. These numbers make us wonder if there's a system that can guarantee a 100% correct decision every time. Often, the evidence presented, and the skills of the lawyers play a big role in the outcome of a case. This leads me to believe that, in general, while we may not completely doubt jury decisions, there may not be a perfect method to always give the right verdict. It ultimately comes down to the word of one person against another, and even with an expert in charge, there will likely be some biases.

 

Moreover, the probability of success based on Condorcet’s theorem has limitations due to the initial assumptions. Is this result realistic and a good model of reality? Maybe not. The assumption of a 0.6 probability for individual jurors represents a simplification made for modeling purposes rather than an accurate reflection of real-world complexity. Individual probabilities of correctness in a jury setting can vary widely based on factors such as legal expertise, personal biases, or prior experiences. Assuming a uniform 0.6 probability for all jurors oversimplifies the intricate dynamics involved in real jury decision-making. The comment on the 0.6 assumption opens the door to future investigations that may delve into the specific characteristics and records of individual jurors, providing a more detailed understanding of the complexities involved in reaching accurate verdicts.

 

After this exploration, even if we have concluded that a jury may not be the best system to decide someone's destiny, we can still determine the optimal jury size to increase accuracy. Considering all the information analyzed and the limitations recognized, a jury comprising 18 members, each with a 60% individual correctness rate, would be the most suitable approach. While using a 70% individual success probability per juror demonstrated the strongest correlation, 0.763, between the probability of success and the number of jurors, setting a probability of correctness beyond 60% is deemed unrealistic. The data at this probability level correlates at 0.634, so the predictability is lower. Nonetheless, it remains sufficient for a reasonably accurate approach. Within the 60% range, the jury sizes with the highest probabilities of reaching the correct verdict in a trial were 9, 18, and 24 members. Considering that the size should not be too large to facilitate decision-making but not too small so that potential individual biases could be diluted, an 18-member jury could meet all the requirements, at least in the proposed model, and likely in a real setting.

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