Bayes Theorem and the Justice SystemNovember 22, 2009

Posted by Stephen Godfrey in Probability.
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I have been reading a previous issue of New Scientist and came across an article by Angela Saini which you can find here on how probability is used in the court room. Also if you goto the page you can take an online test to determine if the article is relevant for you next time you get stuck on jury duty.

It appears that to be a good jury member you need to have a good understanding of conditional and Bayesian Probability. So firstly what do I mean by conditional probability?

Let us consider two events ${A}$ and ${B}$ then the conditional probability of ${A}$ occurring given that ${B}$ has already happened is denoted by ${\mathbf{P}(A|B)}$. Let us consider a fairly simple example. Consider rolling two unique 6 sided dice, let ${A}$ be the even where the sum of the dice is ${8}$ and let ${B}$ be the event where one dice landed on 4. So here we have that ${\mathbf{P}(A)=5/36}$ and ${\mathbf{P}(B)=1/6}$, however here we have that

$\displaystyle \mathbf{P}(A|B)=\frac{1}{6}, \ \ \ \ \ (1)$

because we already know that we have one dice being a 4 so the other dice also needs to be a 4 and that only happens one time in six.

Technical aside: To those probability nerds out there I know I should have labeled these dice, die 1 and die 2 to avoid any complications in finding ${\mathbf{P}(B)}$ however I am just trying to give a simple overview and will just skim over this.

The actual definition of conditional probability is given as follows. If ${\mathbf{P}(B)>0}$, the conditional probability of ${A}$ given ${B}$ is

$\displaystyle \mathbf{P}(A|B)=\frac{\mathbf{P}(AB)}{\mathbf{P}(B)}, \ \ \ \ \ (2)$

where we read ${\mathbf{P}(AB)}$ as the probability of both ${A}$ and ${B}$ occurring at the same time. You can check that we could have used this expression in the above example.

The key point here is that the probability of an even occurring can change if we are given some extra information. Also note that if the two events are independent (i.e. are not related to each other) then ${\mathbf{P}(A|B)=\mathbf{P}(A)}$.

Now what has this got to do with court cases? It has all to do with how some evidence will be presented in court, normally you should consider it as a conditional probability. We have all watched a some tv show that has involved some court case where an expert witness has said that “only 3% of the population has a AB blood type and so does the defendant” So what would we make of this in terms of probability?

Let ${I}$ be the even the defendant is innocent and let ${E}$ be the event that some evidence is being used for or against the defendant. So what out expert witness has given us is ${\mathbf{P}(E|I)=0.03}$. We view it this way any person from that 3% of the population could have left that blood and secondly we assume innocence and have to prove guilt.

What we want to know is ${\mathbf{P}(I|E)}$ so we need some way to relate these two probabilities. This is where Bayes formula comes into the picture.

Let ${I}$ and ${E}$ be the same events as above then

$\displaystyle \mathbf{P}(I|E)=\frac{\mathbf{P}(E|I)\mathbf{P}(I)}{\mathbf{P}(E)}. \ \ \ \ \ (3)$

Here you the jury would have a gut feel for what ${\mathbf{P}(I)}$ should be, for instance motive, past record and even the way he looks. The probability ${\mathbf{P}(E|I)}$ would be given to you by the person that gives the evidence and ${\mathbf{P}(E)}$ can be calculated as

$\displaystyle \begin{array}{ll} \mathbf{P}(E)&= \mathbf{P}(E|I)\mathbf{P}(I)+\mathbf{P}(E|\sim I)\mathbf{P}(\sim I)\\ &=\mathbf{P}(E|I)\mathbf{P}(I)+\mathbf{P}(\sim I), \end{array} \ \ \ \ \ (4)$

where ${\sim}$ stands for not i.e. ${\sim I}$ means not innocent. Also note that ${\mathbf{P}(E|\sim I)=1}$, as here the defendant actually committed the crime. Lets have a look at the example

Suppose that you are 80% certain that the defendant is innocent. So we have ${\mathbf{P}(I)=0.8}$. A Forensics expert gives some evidence that states that some blood of type AB was found at the scene and only 3% of the population have that type of blood and that the defendant has type AB blood. This means that we take ${\mathbf{P}(E|I)=0.03}$. To find ${\mathbf{P}(E)}$ we substitute in to the equation

$\displaystyle \mathbf{P}(E)=\mathbf{P}(E|I)\mathbf{P}(I)+\mathbf{P}(\sim I)=(0.03\times0.8)+0.2=0.224 \ \ \ \ \ (5)$

So once this evidence is given to you the defendants probable innocents plummets from 80% down to 22.4%. In this example you could still not convict using just this pice of evidence. What you can do is keep on adjusting ${\mathbf{P}(I)}$ during the entire trial (where relevant) using the above reasoning.

I will finish this post with four problems in understanding probabilities in court cases are (in no real order):

• 1) Prosecutor’s or Defendant’s Fallacy
• 2) Ultimate Issue Error Explicitly taking a small ${\mathbf{P}(E|I)}$ with the defendants likelihood of innocence.
• 3) Base-Rate Neglect
• 4) Dependent Evidence Fallacy This is related to the independence or dependence of events. In terms of court cases this would pop up in genetic effects. For instance we all know that certain physiological problems run in the family be it breast cancer or disease. If two events are independent from each other then the probability of both of these events happening can be found by multiplying both of the probabilities together.
However using the breast cancer example there is a 1/8 chance of a women having breast cancer during her life. So what is the probability of a mother and daughter both developing cancer during there lives. Well if mother has breast cancer then it is more likely that the daughter could develop breast cancer sometime during her life. I don’t know what that chance is so for sake of argument lets just say that it is twice as likely then the average person, that is a 1/4 chance. So we would find that there is a (1/8)(1/4)=1/32 chance that both mother and daughter will have cancer some time during there lives.

So this begs the question, if every one can be called up for jury duty should we be teaching more probability in schools so every one can understand trials that can include many confusing probabilities?