By Kartik Donepudi, V Form

A Statistical Analysis of Jury Sizes and Guilty Verdicts

Given that a certain percentage, p, of jurors in a trial are inclined to vote guilty, which of the following is more likely?

• 6 jurors ruling guilty in a 6-person jury
• 10 or more jurors ruling guilty in a 12-person jury

1.

Let us define Event A as 6 out of 6 jurors ruling guilty.
Let us define Event B as 10 or more out of 12 jurors ruling guilty.

Both Event A and Event B can be represented with binomial distributions. This is because they are binary events (jurors can rule either guilty or not guilty), we can assume each juror’s decision is independent of the others’, we are interested in the number of jurors who rule guilty, and each juror essentially has the same chance of ruling guilty because a given percentage, p, of jurors are inclined to rule guilty.

The biggest misgiving of this approach is that the jurors’ decisions might not be independent, as jurors can and often do communicate and influence each other.

Event A Binomial Information:

• Number of trials: 6
• Chance of success: p
• Target number of successes: 6

Event B Binomial Information:

• Number of trials: 12
• Chance of success: p
• Target number of successes: 10, 11, or 12

2.

With a probability, p, of 0.6:

P(A) = binompdf(6, 0.6, 6) = 0.0467

P(B) = (1 – binomcdf(12, 0.6, 9)) = 0.0834

The probability of Event B is greater, meaning it is more likely that a jury of 12 will convict with a 10-2, 11-1, or 12-0 decision than a jury of 6 convicting with a 6-0 decision.

With a probability of 0.8:

P(A) = binompdf(6, 0.8, 6) = 0.262

P(B) = (1 – binomcdf(12, 0.8, 9)) = 0.558

The probability of Event B is greater, meaning it is more likely that a jury of 12 will convict with a 10-2, 11-1, or 12-0 decision than a jury of 6 convicting with a 6-0 decision.

With a probability of 0.9:

P(A) = binompdf(6, 0.9, 6) = 0.531

P(B) = (1 – binomcdf(12, 0.9, 9)) = 0.889

The probability of Event B is greater, meaning it is more likely that a jury of 12 will convict with a 10-2, 11-1, or 12-0 decision than a jury of 6 convicting with a 6-0 decision.

3.

If p = x and P(A) = y on a Cartesian plane, the graph of P(A) looks like:

And the graph of P(B) looks like:

In the above graphs, the dotted black lines represent the domain and range limits of each function. The input and output are both probabilities and are therefore between 0 and 1.

4.

If we graph the probability of both events together, it looks like:

The green line is P(A) and the purple line is P(B). Their binomial probability functions are:

If we zoom in further, we can see that the two functions intersect at p = 0.4592:

From the graph, it is evident that P(A) is greater than P(B) left of p = 0.4592, but P(B) is greater than P(A) right of p = 0.4592. 

As such, when less than 45.92% of the jury is inclined to vote guilty, a defendant is more likely to be convicted by a jury of 6 with a 6-0 decision than he is by a jury of 12 with a 10-2, 11-1, or 12-0 decision. In this range though, the probability of conviction is extremely low(less than 0.9373%).

When more than 45.92% of the jury is inclined to vote guilty, a defendant is more likely to be convicted by a jury of 12 in a 10-2, 11-1, or 12-0 decision than he is by a jury of 6 with a 6-0 decision. This range of conviction probabilities includes values from 0.9373% upwards, meaning this scenario will likely be the more common type of conviction. 

While Event A is more likely when less than 45.92% of the jury is inclined to vote guilty, it is unlikely that these trials will actually end in convictions. Also, the difference between the probability of Event A and Event B in this range is significantly smaller than the difference in probabilities when more than 45.92% of the jury is inclined to vote guilty, a fact visible in the graphs above. 

Part of the reason that larger percentages of guilty-voting jurors cause 10-2, 11-1, and 12-0 decisions to have higher probabilities is that the binomial function for these decisions contains x terms raised to the 10th, 11th, and 12th powers, whereas the 6-0 decision’s binomial function contains an x raised to only the 6th power. This means that the 10-2, 11-1, and 12-0 function increases faster than the 6-0 function and eventually surpasses it, something we see in the graph at p = 0.4592.

In conclusion, a defendant is more likely to be convicted with a 6-0 decision than with a 10-2, 11-1, or 12-0 decision when less than 45.92% of the jury is inclined to vote guilty. When more than 45.92% of the jury is inclined to vote guilty, a defendant is more likely to be convicted with a 10-2, 11-1, or 12-0 decision than with a 6-0 decision.