Statistics in Court. Lesson 1: Beware multiplying probabilities.

April 24, 2019 By Alan Salzberg

The chances that a randomly selected man will be 7 feet tall or more is about one in a million.  So it follows that the chance of two randomly selected men being 7 footers is the square of one in a million, which is one in a trillion, right?

If we apply this probability to every family in the world (and assume that for every four people there are two parents and two adult sons), the chance that any family will have two seven foot sons is still quite small – there is only about a 1 in 500 chance that there’s even a single family on Earth with two sons that are seven feet tall.

Then how do you explain Brook and Robin Lopez and Marc and Pau Gasol, two pairs of 7 feet tall brothers who play basketball in the NBA? 

The problem is the assumption that we can compute the chances of two seven footers simply by squaring the chances that one person will be seven feet tall.  That squaring only works if the two heights are independent.  The height of two brothers is not statistically independent because the similar genes of the two brothers mean that the heights are likely to be closer than two randomly chosen men (even if the brothers are adopted, they are likely to be more similar in height than randomly chosen men because of the presumably similar diet they had growing up).

Some quick math.  Suppose A is the event that one brother is seven feet tall and B is the event that the second brother is seven feet tall.  The the chances of A and B are: the probability that A occurs multiplied by: the probability of B occurs given that A occurs.  In shorthand, this is written: P(A and B)=P(A)*P(B|A)

P(A and B) only equals P(A)*P(B) when P(B|A)=P(B), and in that case A and B are statistically "independent."  Theoretically, P(B|A) can be anything between 0 and 1.  If A and B tend to occur together, though, then P(B|A)>P(B).  This means that simply multiplying P(A) by P(B) will give you a probability that is too small.

Thus, instead of 1 in a trillion, the chances that two randomly selected brothers are 7 feet may reasonably be anything between one in a million and one in a trillion.  In the case of the Lopez twins, their chances of both being 7 feet is more like one in a million, because they are identical twins and are quite likely to be the same height.    

This all seems pretty obvious in a general sense.  That is, everyone knows that siblings (whether or not they are identical twins) share traits and therefore while you might be surprised that one sibling has some unique trait, you would be much less surprised when the other sibling has that same trait. 

So it is shocking that a court and jury accepted this naive and clearly false assumption regarding two babies with the same parents.  But they did, and they did so with dire consequences.   

Sally Clark was falsely convicted of killing her infant son due in part to a expert’s claim that the probability that the child’s death occurred by accident was 1 in 73 million (the latest issue of Statistical Significance recounts this case and other cases where probabilities were misused in court in  Beyond highlighting the Sally Clark case, this article also highlights the general issue with misuse of the multiplication rule and the implied independence assumption). 

How did the expert get to 1 in 73 million?  The expert took the chances of a sudden infant death by accident at one in 8,543 and squared it.  As explained above, the true chances might be as low as 1 in 8,543 and would only be one in 73 million if the siblings chances of death were independent.  While that 1 in 8,543 still seems low, it is not when you consider there are about 700,000 births a year in the U.K. (where Sally Clark resided).  This implies there will be about 80 such deaths a year.  Sadly, the poor probability calculation and the inevitability of some such deaths was not brought up until appeal.  Sally Clark sat in jail for three years before the case was reversed.  Lest you think she was convicted for reasons other than the incorrect probability calculation, the Statistical Significance article notes that her arrest was on scant evidence and provides a quote from one of the jurors: “whatever you say about Sally Clark, you can’t get around the 1 in 73 million figure.”