## Justice is blind

2010-02-15

There can be no denying that chance is a part of justice. After all that is why all statues that represent it are blind. And (un)fortunately it will probably stay that way as long as humans are part of it. Also jury trial is probably as old as the concept of justice itself. What it means, in the current times, is that a group of people is chosen to decide questions of fact in a trial. What I’m interested in this post is to see how the bias of the general populace can influence jury decisions.

In the modern justice systems there is also this concept called presumption of innocence. It is a very nice concept in theory but as usual in practice sometimes it’s not easy to follow it. Sometimes people just assume that if you’re on trial you must be guilty. Sometimes the trial involves powerful emotions an it’s easier to follow them then try to reason logically. As I said the jury is made of people, and people have their biases.

Let’s suppose the following scenario: there are N people in a jury, and they have to decide if the defendant is guilty or not. But the trial was not extremely conclusive (which is normal, because if the verdict would be obvious this post would have no purpose). So either the lawyers were really good (or really bad) or the evidence was not strong enough for any side, but the jury members have to basically give a guilty/not guilty verdict by chance. Of course each member of the jury will use its own rationalization to justify the answer that (s)he gave but I’m aiming for a probability problem here, not a sociology one.

In most cases a jury needs unanimity to give a verdict. So let’s add some more constraints to our problem. Let’s suppose that if the majority of the jury will vote one way (guilty or not guilty) they will convince the rest to vote the same way. Also if the guilty/not guilty votes are equal the jury will vote again (randomly) until we have majority in which case we apply the first rule.

So we can have multiple results of the jury vote depending on the number of members in the jury. If N is even there are 3 possible resulting scenarios:

• the defendant goes to jail if more than $\frac{N}{2}$ members of the jury vote that he’s guilty
• the defendant goes free if less than $\frac{N}{2}$ members of the jury vote that he’s guilty
• the jury will vote again if the votes are split equally. This scenario is here just for completeness since the jury will have to vote again

If N is odd we have only the first two. From now on I will consider N = 12 for the purpose of getting some actual calculations done.

### All jury members are NOT biased

In this case each jury member will give the guilty/not guilty verdict with probability of 0.5. So the defendant has equal chances to end up in jail or free, which makes sense since the trial was inconclusive and there is no way to take a clear decision.

### All jury members are identically biased

In this case each jury member will give the guilty/not guilty verdict with probability different from 0.5. An easy model for this is to imagine we have a big bowl with black and white balls, and the probability to extract a black ball (guilty) is equal to the bias of the jury. We then proceed and extract N times (with replacement) from that bowl.

The probabilities corresponding to the 3 possible scenarios described earlier are:

$p(\text{jail}) = \sum_{i=\left( \frac{N}{2}+1 \right)}^{N} C(N, i) \cdot p^i \cdot (1-p)^{(N-i)}$

$p(\text{free}) = \sum_{i=0}^{\left( \frac{N}{2}-1 \right)} C(N, i) \cdot p^i \cdot (1-p)^{(N-i)}$

$p(\text{re-vote}) = C\left(N, \frac{N}{2}\right) \cdot p^{\frac{N}{2}} \cdot (1-p)^{\frac{N}{2}}$

### Only ONE jury member is biased

In this case each jury member will give the guilty/not guilty verdict with probability of 0.5 except one of them which is biased. An easy model for this is to imagine we have two big bowls with black and white balls, one with an equal number of black and white balls from which we extract (with replacement) N-1 times and one from which we extract once where the probability to extract a black ball (guilty) is equal to the bias of the single jury member.

We can organize the possible outcomes in the following table (just to make it easy to see what terms we need to add):

 1-p(bias) p(bias) C(11, 0)0.511 free free C(11, 1)0.511 free free C(11, 2)0.511 free free C(11, 3)0.511 free free C(11, 4)0.511 free free C(11, 5)0.511 free re-vote C(11, 6)0.511 re-vote guilty C(11, 7)0.511 guilty guilty C(11, 8)0.511 guilty guilty C(11, 9)0.511 guilty guilty C(11, 10)0.511 guilty guilty C(11, 11)0.511 guilty guilty

The terms in the first column are the probabilities for possible outcomes of the (N-1) judge members who have no bias. The first row shows how the biased single member can vote. To get the probabilities of each possible scenario just add all the products as described in the table.