MAth on Trial (non-fiction)

Subject: Mathematics

Type: Non-fiction

Authors: Coralie Colmez and Leila Schneps

Math on Trial front cover

How long is it?

There are ten main sections and 274 pages. It typically takes 6 hours and 11 minutes to read.
 

Is it easy to understand?

All mathematical concepts are well-explained with analogies and examples.
 

Who is it for?

The book is aimed at a popular audience. It is suitable for teenagers, although it includes mention of violence which make it less suitable for younger children.
 

How recent is it?

Basic Books published it in 2013.
 
 

What to expect

The book describes ten legal cases in which mathematics has been misused and misunderstood.
Math Error 1: multiplying non-independent probabilities

In order to measure the probability that several events will occur, the separate probabilities of each event should be multiplied together. However, this only works when the events are independent. If you multiply the probabilities of events that are not independent of each other, the answer you get will be significantly smaller than is accurate.

The Case of Sally Clark: Motherhood Under Attack
In this case, Sally Clark was accused of murdering her children. In order to argue that the probability of two of her children having died by chance was miniscule, Roy Meadow multiplied  the probabilities of each death occurring as a result of  SIDS (crib death), despite the events not being completely independent. There are risk factors for SIDS, which means that it is not a purely random event. 
Math Error 2: unjustified estimates
The effect of an estimation error is frequently overlooked, as though the only important thing is having a number with which to back up an argument. 
 
 
 
The Case of Janet Collins: Hairstyle Probability
In this case, a couple was accused of theft because they mostly matched the descriptions of the victim and witness. The prosecutors attempted to provide the jury with an estimated likelihood of another pair in the neighbourhood being equally similar to provided description, however, they did so using arbitrary guesses of non-independent probabilities.
Math Error 3: trying to get something from nothing
Here, non-independent probabilities were multiplied, unjustified statistical estimates were made and on top of that, the most important of these estimates was based on a finding of zero in the examined sample.
 
The Case of Joe Sneed: Absent from the Phone Book
In this case, it was found that a Robert Crosset had purchased the murder weapon, yet no such name was found in the phone books. The prosecuta estimates the frequency of this name in the general population, but in doing so, assumes that the name was not made up, despite his trying to prove that Joe Sneed was Robert Crosset. In addition, he assumes that the name is equally distributed in the united states and that its frequency can be estimated using a set of phone books.
Math Error 4: double experiment

The judge made the mistake of assuming that a new DNA test on a presumed murder weapon would provide no more information than the first one, rejecting a second test that might have uncovered the truth.

 

The Case of Meredith Kercher: The Test That Wasn’t Done
The judge said that the fact of having run an experiment independently twice and obtaining the same result both times does not increase the reliability of said result. This shows a complete misunderstanding of probability.
Math Error 5: the birthday problem

How many people do you have to put in a room for there to be a 50-50 chance that two of them share the same birthday? The answer is about 23. How many people do you need to put in a room for there to be a 50-50 chance that one of them was born on the 1st of January? In this case, the answer is 253. These two answers seem counterintuitive to most people, who might have guessed that 183 (roughly half of 365) was the answer to either of them. It is also confusing because the two very different answers are to two questions that sound similar.

The Case of Diana Sylvester: Cold Hit Analysis
In this rape and murder case, Bicka Barlow attempted to show that the probability of two potential murderers sharing five and a half of a sample’s loci (genetic pairs) was relatively high, so that it cannot be assumed that the only found match is the culprit. However, instead of finding the chance of a match with the sample in question, she found the chance of a match with any other person. She confused a match between any two people with a match to one particular DNA sample.
Math Error 6: Simpson’s Paradox

Here is an example of Simpson’s Paradox: the average score of every ethnic group of students on a standardised test administered yearly has increased, yet the overall average has remained the same. This is possible when the distribution of the population within the ethnic groups changes.

The Berkeley Sex Bias Case: Discrimination Detection
The percentages of male and female applicants admitted to Berkeley apparently indicated sex bias, however, with more investigation, it was found that in individual departments, a larger proportion of female applicants were accepted. The reason for this was identified as a large difference in the number of males and females applying to certain departments, such as mathematics and engineering.
Math Error 7: the incredible coincidence
It can be misleading to calculate the probability in retrospect of an event that has already occurred. If an event’s likelihood was small enough, suspicion may be aroused.



The Case of  Lucia de Berk: Carer or Killer?
In calculating the probability that Lucia de Berk, a nurse in the Netherlands, had been present at so many of the deaths and near-misses of her patients purely as a coincidence, the prosecution made many mistakes. This was because the prosecutors made their calculation from an inaccurate table and they also multiplied non-independent probabilities.
Math Error 8: underestimation
If you start with the number 1 and double it eighty times, you are left with an extraordinary number. It is important to recognise that an exponential growth pattern starts off relatively slowly, but later accelerates dramatically. The number you will have after doubling 1 many times is easy to underestimate.
The Case of Charles Ponzi: American Dream, American Scheme
Charles Ponzi set up a scheme in which he promised to double each investor’s money every 90 days, but the way he was profiting from buying and selling stamps was illegal. Soon he was unable to continue making profit, so he simply handed money from new investors to the ones who wanted out. Ponzi and his investors underestimated the amount of money and investors Ponzi would need in order to continue doubling every investment four times per year.
Math Error 9: choosing a wrong model

The simpler a mathematical model, the less likely it is to accurately apply to a real-life situation. In fact, applying any mathematical model to real life is risky, let alone in a courtroom.

 

 

The Case of Hetty Green: A Battle of Wills
The binomial formula was used in this case to approximate the numbers in a table. However, the numbers were not as similar as they seemed to be. This meant that the prosecutor’s calculation of the probability that two signatures were identical by chance was skewed.
Math Error 10: mathematical madness

When calculating probability of an event occurring, it would be a mistake to disregard the number of opportunities in which it could have occurred. It’s the difference between saying you scored 2 out of 2 penalty kicks and saying you scored 2 out of 100.

The Dreyfus Affair: Spy or Scapegoat?
Bertillon tried to prove that a document contained a secret code, by investigating the positions of letters on the page. He found 8 pairs of letters in identical positions relative to some vertical lines on the page and concluded that there was a miniscule chance of this happening unintentionally. But this was because he had worked out the probability of 8 pairs out of 8 being in identical positions, rather than 8 or 9 or 10 or 11 and so on… out of 26.

My thoughts…

Should Mathematics have a role in the detection and proof of crime?

I think that DNA analysis is a valuable tool in the courtroom and therefore, so is mathematics. This is because we use probability to help us determine the implications of matching DNA samples. When the DNA sample is degraded, incomplete or contaminated, inaccuracies can arise in our probability calculations, but ignoring the DNA altogether is not an option. This means that as long as there is forensic analysis, mathematics will have a role in the detection and proof of crime and will be useful most of the time. Thankfully, awareness about the controversy over mathematics in court is increasing, so more people are suggesting ways to prevent further mathematical misjudgements like the ones above.

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