Before I explain what the prosecutor’s fallacy is, here’s a real life example where it affected (for the worse) the life of someone.
Sally Clark was a British solicitor, and in 1996 his first son died, only weeks after his birth. In 1998 the same pattern occurred, when his second son died only weeks after his birth. Clark was tried for the murder of her two sons, and the prosecutor used the following argument:
“According to an expert pediatrician, the chance of a newborn dying in a similar circumstance is 1 in 8500. So the chance of two newborns of the same mother dying in this same manner is 1/8500 * 1/8500, which is equal to 1 in 72,250,000. In other words, this event is extremely unlikely and therefore Sally Clark must be guilty of murder.”
The argument seemed plausible, and Clark was convicted and went to jail. She served over one year before people started questioning the statistics involved in her trial. In 2001 her process was revised and she was released.
Now here’s why the calculation above was not correct.
First, you would be able to multiply the individual probabilities together only if the events were statistically independent of each other, but most health issues of sons of the same mother are not independent. So the real probability would be much lower than 1-in-72 million.
Second, many people in the trial (including the jury, probability) fell victim to the “prosecutor’s fallacy”. That is, people started assuming, and some news outlets started reporting, that the 1-in-72 million was the probability of Clark being innocent.
The Prosecutor’s Fallacy In Clark’s Case
I already mentioned that the 1-in-72 million number is probability not correct. But let’s assume it is. Even in that case that number had nothing to do with the probability of Sally being innocent, as people wrongly assumed.
The 1-in-72 million number refers to the probability of two children dying of SIDS (Sudden infant death syndrome). In order to find the probability of Sally being guilty or innocent we need to consider the other possible explanations for the death of her children, and their respective probability.
Let’s assume the only other probable case was the double murder. In 2002 a mathematics professor calculated that the probability of a mother murdering both her newborns is 1-in-300 million.
In other words, even with the 1-in-72 million number the children were around 4 times more likely to have died of SIDS then murdered by their mother.
Another way to put it: the chance of the two newborns dying of SIDS was very small, but the chance of a double murder by the mother was even smaller, so her probability of being innocent was considerable.
The appeal court considered those points and decided to revert Clark’s conviction. It also decided to bring several other cases where mothers were accused of murder to revision.
Example with Numbers
Suppose that you in the jury of a trial. The defendant is being accused of murder. The main evidence presented by the prosecution is that the blood type of the defendant matches the one found in the crime scene, and that only 10% of the population has this blood type. According to the prosecutor, this number reveals that the defendant is 90% likely to be guilty.
Do you agree?
Well, you shouldn’t.
Let’s define G as being the event that the defendant is guilty. First of all we need to define the prior probability of G. Prior means without any information about the matter, so one way to do this is to use the statistics of all trials. Let’s say that of all murder trials the defendant is found to be guilty on 30% of them, so initially:
P(G) = 0.3
Now let’s say B is the event of picking a person at random and that person having that same blood type of the crime scene. P(B) is 1/10 therefore (i.e., only 10% of the population has this blood type).
P(B) = 0.1
This means that if picked a person at random and compared her blood with the crime scene it would have 10% chances of matching. If we picked the murderer the chances of a match would be higher. Let’s say they would be twice as much, so 20% (we are not going higher because there are many scenarios where even if we picked the murdered the blood type wouldn’t match). This means that:
P(B|G) = 0.2
This is the probability of the blood matching GIVEN the defendant is guilty. What we want to find, however, is the other way around. That is, we want to find the probability of the defendant being guilty GIVEN the blood type matched. These are different things, and that’s the confusion the prosecutor made (deliberately or not).
According to Bayes’ Theorem:
P(G|B) = P(B|G) P(G) / P(B) P(G|B) = (0.2) (0.3) / (0.1) P(G|B) = 0.6
As you can see the probability of the being guilty GIVEN there was a blood match turns out to be 60% in our case. A significant number, but certainly not as high as 90% as the prosecutor wanted you to believe. And now the question: is 60% probability enough for you to send someone to jail? Well, you decide!