Entering

CASE 01 The Charles Ponzi Case
The Truth About the American Dream PLM Scam
Mathematical error 1.
How to Double Your Profits Every Three Months

CASE 02 Gender Discrimination Case in Graduate School Admissions Exams
UC Berkeley sex discrimination case
Mathematical error 2.
Why the overall average doesn't change even if individual groups' scores increase

CASE 03 The Lucia Duberg Case
How a Nurse Became a Murderer
Mathematical error 3.
The probability of an unbelievable series of coincidences occurring one after another

CASE 04 The Amanda Knox Case
Why DNA Testing Couldn't Catch the Criminal
Mathematical error 4.
How to increase the reliability of probability experiments

CASE 05: The Diana Sylvester Case
The culprit was found using DNA preserved for 30 years.
Mathematical error 5.
Probability of there being someone with the same birthday

CASE 06 The Sally Clark Case
Why a mother became a murderer who killed her child
Mathematical error 6.
When the first sibling dies, the probability that the second sibling will die in succession

CASE 07 The Collins Case
A couple arrested due to the possibility that they share the same description as the criminal
Mathematical error 7.
Estimation of illogical probabilities

CASE 08 The Joe Snead Case
How the son who murdered his parents tried to escape investigation
Mathematical error 8.
The probability that a name not in the phone book actually exists

CASE 09: The Hetty Green Case
The inheritance dispute of the wealthiest woman in American history
Mathematical error 9.
The probability that two signatures match almost perfectly

CASE 10 The Alfred Dreyfus Affair
The Truth About the Dreyfus Affair That Divided 20th-Century France
Mathematical error 10.
Madness expressed in mathematics

In conclusion
References

Can math help us identify the culprit in a murder case?
Math reveals a huge mistake made in court!

Mathematics is a discipline that provides clearer and more definite answers than any other discipline.
So, can these properties of mathematics be applied in court? For example, can mathematics be used to identify criminals? This book explores the efforts of investigators, lawyers, and mathematicians to use mathematics to identify criminals, from the simple handwriting analysis used in the late 19th and 20th centuries to gender discrimination in college admissions, the pitfalls of pyramid schemes, and the accuracy of DNA analysis used in criminal cases today.
However, despite the accuracy and clarity that mathematics provides, it did not always provide the right answer in trials.
Sometimes, instead of identifying the real culprit, mathematics has pointed the finger at the wrong person or acquitted someone who was almost certainly the culprit.
This book tells the stories of people whose lives were ruined by extremely unfair judgments resulting from simple mathematical errors such as miscalculations, misunderstandings of calculation results, or overlooking necessary calculations.
Authors Leila Schnepps and Coralie Colmez meticulously researched the case and interviewed those involved, detailing the events leading up to the incident, the police investigation, the social repercussions, and the proceedings leading up to the trial.
Through these true-to-life stories, you'll learn that mathematics can truly be a matter of life and death, and that even mathematics, which allows no margin for error, can have fatal consequences in the wrong hands.

-- Historically, it has been very rare for mathematics to come to the forefront in criminal cases.
Even in such cases, most of them only go so far as to find out the probability that the identification results that have already been performed are correct.
Since this trend is consistent across both public and private law, one might wonder why mathematics is rarely used in trials.
Since trials are where mathematical errors, which are prevalent in any field, are most clearly revealed, we thought it worth collecting and examining relevant cases.
Because trials are a great way to show how faulty reasoning can actually lead to serious consequences.
(syncopation)
Although mathematics can sometimes lead to completely wrong decisions, the main argument of this book is that probability is not a useless tool in the courtroom.
(Omitted) There is no doubt that mathematics is a useful tool, and given that DNA is widely accepted as evidence in criminal trials today, I believe that mathematical analysis will definitely have to be included in criminal trials in the future.
But to do that, we need to be confident that no mathematical errors occur in the trial, and to do that, we need to look at the errors that actually occurred.
_6~7p

Why do people fall for multi-level marketing scams?
Analyzing Charles Ponzi's Scam That Turned 20th-Century Boston Upside Down

In 2009, the largest Ponzi scheme in history occurred in the United States, and its mastermind, Bernard Madoff, was arrested and sentenced to 150 years in prison.
The total losses resulting from Madoff's fraud amounted to $18 billion.
Even at the current exchange rate, this represents a loss of over 21 trillion won.
What on earth was a Ponzi scheme that made people fall for Madoff's scheme so easily?
A Ponzi scheme is a type of investment fraud that uses money from new investors to pay profits to existing investors.
Simply put, it is a multi-level marketing scam, and it is named after Charles Ponzi, a fraudster who used this method on a large scale in the United States.
Ponzi, an Italian immigrant, started a business that promised to double his investors' profits in just 90 days.
But if you do a little math, you'll see that no business can double its money in three months.
In other words, Ponzi did not run the business with the investors' money, but committed fraud by taking money from new investors and paying it to existing investors.
This case is so famous that it shook up Boston, where Charles Ponzi's business was based in the early 20th century, and almost the entire United States.
However, similar pyramid schemes like the Madoff case still occur in modern times because there are still people who are infinitely vulnerable to mathematical concepts.
In this chapter, we will mathematically examine the business models of multi-level marketing scams and explore how to avoid them.

-- In modern times, even if you make a profit from an investment in a Ponzi scheme, you can suffer even greater losses if a lawsuit is filed.
Why is that? Because any Ponzi scheme (or rather, a scam) can fail very quickly.
In 1920, no one could have predicted this.
The power of the dream of becoming rich was so great that 2,010 people fell for this trick.
The disaster began only after the incident had occurred.
If you're tempted by investment products promising eye-popping returns, you should do some math before deciding whether to invest.
_27p

-- As the recent Bernie Madoff case (the largest Ponzi scheme in U.S. history, sentenced to 150 years in prison in 2009 - translator's note) illustrates, pyramid schemes wield a powerful force on greedy but oblivious investors.
And not just one or two people, but thousands, tens of thousands of people.
If there were no people who fell for it, multi-level marketing scams would have disappeared long ago.
Why don't people learn from Ponzi's legendary scam? _37-38

Can gender discrimination be mathematically proven?
How to Spot the Illusions of Statistics

In 1986, UC Berkeley was sued for allegedly discriminating against female students in graduate school admissions.
The passing rate for female students was extremely low compared to that for male students.
When the issue was raised, a committee was formed to investigate it, and they tried to mathematically confirm the existence of gender discrimination by calculating the number of applicants and successful applicants.
However, upon closer investigation, something strange was discovered.
When looking at the male and female acceptance rates for each department, there were no particular problems, and in fact, more female students were accepted. However, when looking at the statistics for the entire school, the acceptance rate for female students was much lower.
This phenomenon, called Simpson's paradox in statistics, occurs when important information is omitted or ignored in statistics.
In this case, the problem arose because the 'ratio of male and female students applying to the department with the highest acceptance rate' was not taken into account.
Because the gender ratio of applicants varies by department, when looking at each department, the acceptance rate for female students appears to be much lower overall, even though there are more female students accepted.
Human intuition often leads to conclusions that are not true, but sometimes even conclusions based on statistical data are not as clear-cut as conventional wisdom would suggest.
The authors of this book argue that to avoid such misjudgments, we should not simply accept given numbers, but rather develop the mathematical ability to closely examine and analyze the process.

-- The average SAT reading score in 2002 was the same as in 1981.
However, the racial scores classified by the evaluation committee showed that during the same period, scores by whites rose by 8 points, blacks by 19 points, Asians by 27 points, Puerto Ricans by 18 points, and American Indians by 8 points.
How can each group's scores improve while the overall average remains unchanged for 21 years?
This remarkable case is a classic example of a phenomenon known as Simpson's paradox.
Over the past two decades, average scores on standardized tests administered annually have steadily increased across all racial groups of test takers.
However, the overall average remains unchanged.
(Omitted) How is it possible that all groups improved their performance, yet the overall score remained unchanged? The secret lies in a factor that doesn't appear on the chart, but plays a crucial role.
In this case, the factor is the overall change in the population of each racial group.
_42~43p

The Mathematical Trap That Turned a Devoted Guardian into a Murderer
Why did a kind nurse and devoted mother become a murderer?

Beyond large-scale multi-level marketing scams that cause financial losses and gender discrimination, there are also more direct instances where mathematics has taken away individual freedom.
The statistics and probabilities used in court sent innocent people to prison.
In 2001, Dutch nurse Lucia Derberg was charged and convicted of 13 counts of murder and four counts of attempted murder.
According to hospital officials, they were present at the scene of patient deaths far too often to be considered a coincidence.
The hospital director, who had counted the number of times Lucia was present when a patient was in critical condition, reported Lucia Duberg to the police based on these statistics.
According to his calculations, the odds of Lucia killing that many patients were 1 in 342 million, a number far greater than the number of nurses in the world.
In other words, the probability that the patient's death occurred naturally was too low.
In a similar case, in 1996 in England, a mother was convicted of murdering two children.
The first and second children died within a year of each other, and the odds of two children dying consecutively for unknown reasons were only 1 in 73 million.

-- Neither the table nor the calculations were entirely accurate, but looking at their contents, it is not difficult to understand why Henk Elfers and Paul Smitz were so convinced that Lucia was guilty.
During the nine months that Lucia worked at Juliana Children's Hospital, there were 1,029 nursing shifts, and she worked 142 of them.
During Lucia's shift, seven incidents occurred that the hospital reclassified as "unnatural deaths."
(syncopation)
It is a suspicious and surprising number to anyone who looks at it.
There is no doubt that Lucia faced these situations far more frequently than the average nurse faces in her career.
_76~77p

"So Harry's chances of dying from SIDS after birth are the same as Christopher's? One in 8,543? Like flipping a coin, with the same odds? Does it always come up heads or tails?" asked Sally's lawyer.
“It’s the same odds that people bet on in the Grand National,” Meadow said quietly, dryly.
“Let’s say you bet on a horse that was predicted to win with a 1 in 80 chance last year and then this year you bet on a different horse that was predicted to win with the same chance and you win again.
A 1 in 73 million chance is like betting on a horse with 1 in 80 odds to win four years in a row.
The same goes for the sudden infant death mortality rate.” _190-191p

How could mathematics have come to such a wrong conclusion?
Math reveals fatal mistake made in court!

But there was a huge error in this probability calculation that made the nurse and the mother murderers.
When compiling statistics on the deaths of patients that led to the claim that Nurse Lucia had intentionally killed patients, the hospital compiled a list of suspicious cases and checked to see if Lucia had been present in each one.
However, the death of a patient in a hospital cannot be considered a 'suspicious' event in itself.
Because in a hospital where patients gather, patients in serious condition can die at any time.
In other words, the list of suspicious cases claimed by the hospital was ultimately information collected posthumously, after Lucia was suspected.
In other words, it is not that Lucia was present in all the suspicious cases, but rather that they were viewed as suspicious cases because Lucia was present.
There was also a major mathematical error in a British case where a mother was suspected of infanticide.
How did the prosecution arrive at the "1 in 73 million" figure? Based on a report prepared by the British Ministry of Health at the time, the prosecution calculated the probability of a child dying accidentally to be 1 in 8,543. Since the two children died, they considered each incident independent and simply squared the probability.
However, sudden infant deaths can occur by chance, but they can also be caused by other factors that are not yet known.
Especially if the same case occurs twice in the same household, genetic factors should also be suspected.
The prosecutor did not consider this possibility and framed the mother as the culprit.
In fact, a later review of hospital records revealed that the child had not died of an unknown cause, but rather had been infected with a bacteria.

-- If you were to make a table listing the number of deaths every nurse in any country has encountered, someone unlucky enough to have experienced a disproportionately high number of deaths would be at the top of the list.
So, should we arrest that person? The purpose of the calculation is to determine whether this person falls within the natural statistical distribution—whether he or she is a murderer or not.
_77p

-- Eight of the 5,000 children born to families after the death of a child from sudden infant death syndrome died, according to records from the Infant Care Program.
This statistic is so high compared to the 1 in 73 million (8/5,000 = 1 in 400 - translator's note) that if Meadow's figures are correct, the sudden deaths of two children in one household should only happen once every 100 years in the UK.
However, statistics from the Early Childhood Care Programme show that such tragic incidents occur every few years in the UK.
In fact, many couples who had lost two or even three children to SIDS sent letters of encouragement to the Clarks.
_189p

Is mathematics also used in handwriting analysis?
Why did a mathematician need to be involved in uncovering the case of forged wills and leaks of state secrets?

The Dreyfus Affair, which nearly split 19th-century France politically, is known as a prime example of the state suppressing individual freedom, but few people realize that mathematics was deeply involved behind it.
Alfred Dreyfus, a French officer who was accused of being a German spy, was accused of having handwriting that closely resembled that of a confidential memo found in an office trash can.
At this time, the mathematician who was called as a witness by the prosecution calculated the probability that the handwriting in the note and Dreyfus' handwriting could be so similar by chance in court, and argued that since the probability was very low, Dreyfus had to be the owner of the note.
Dreyfus was ultimately framed as the culprit and was forced to spend nearly five years in exile on a deserted island.
Hetty Green, the wealthiest woman in 20th-century America, was also suspected of forging her aunt's will in her youth.
It was claimed that the signature on the aunt's will was so similar to signatures on other documents that it was suspected that Hetty Green, the will's primary beneficiary, had copied the signature.
Here too, mathematicians collected Hetty's aunt's signatures, compared them, and compiled statistics, proving that Hetty had forged the will, as no two human signatures could be so perfectly identical.
Although historically Alfred Dreyfus has been proven to have been falsely accused, the question of whether Hetty Green forged her will remains a mystery.
This is precisely why mathematics has been underused in courts for decades.
In these cases where mathematics came to the forefront, mathematics did not always produce the right answer or a definitive solution.
In some cases, mathematics has made such a huge mistake that it has pointed the finger at the wrong person, and in some cases, it has ruined the life of an individual.

-- In court, Benjamin Peirce said the odds were about 1 in 530, which is a very small number.
He explained to the court that this value was 1/2,666 of a millionth of a millionth of a millionth (this is 1/2,666 × 1018, which is about three times larger than it actually is, but keep in mind that there were no calculators in the 19th century).
In any case, the 1/530th value he mentioned was significant.
“A number that is beyond human senses (…) such an impossible number means that it is also impossible in reality.
Such mirage-like probabilities do not exist in reality.
“It goes beyond the realm of numbers that the law must consider to an unimaginable degree.”
In other words, the probability that two signatures would coincidentally be so perfectly identical that they would be indistinguishable is negligible, and therefore, if two signatures are so identical that one is forged, then it is likely that the other is forged.
So, who else but Hetty, who would have benefited from the will, would have forged the signature? _275~276p

Is it right to use mathematics in court?
A true-to-life drama about a court that suppresses everything from property to personal freedom!

The precedents of mistrials and miscarriages of justice surrounding mathematical errors in court led to a tendency in the American legal community for a time to avoid using mathematical concepts in determining guilt or innocence.
However, with the recent advancement of science and technology, DNA analysis is often used to identify criminals.
This book also introduces cases that arose surrounding DNA analysis, and introduces cases where mathematics was used to identify criminals.
Through the arguments between the plaintiff and the defendant regarding the DNA analysis results presented as evidence in the 2007 murder of a British student in Perugia, Italy, and the DNA analysis results that led to the arrest of the perpetrator in a case that had remained unsolved for over 30 years, readers can understand why DNA analysis, which seems to be infinitely accurate and clear, is controversial.
In the case of the murder of a British student that shook the United States and Europe, a suspect who was extremely close to being guilty was found not guilty because a judge questioned the reliability of DNA testing, and the lawyer of a suspect arrested as the culprit in an unsolved case lost the case because he brought up an incorrect probability theory.
In the mid-20th century, there was even a case where a suspect was determined to be the criminal by calculating the probability that a random person would have the same appearance as the criminal.
A woman was mugged, and the mugger was a white woman with blonde hair and accompanied by a black man. So, the probability that a person is blonde, white, and married to a black man was calculated and multiplied together.
The suspect identified as the culprit was convicted and served time in prison even though his description did not perfectly match the eyewitness accounts.
In this book, authors Leila Schneps and Coralie Colmez warn that even mathematics, which seems like a useful and clear-cut discipline, can lead to erroneous conclusions in the wrong hands.
However, mathematics cannot be completely banished from the courts.
As the science and technology used in crime analysis advances, mathematics will become increasingly important.
Therefore, the two authors argue that not only lawyers but also the general public who will one day sit in the jury box must be equipped with mathematical thinking.
If you have the basic mathematical knowledge and critical eye to closely follow mathematical theory and computational processes, you will be able to correct errors and use mathematics usefully.

-- As the DNA-related cases discussed in this book (the Meredith Kercher and Diana Sylvester murders) demonstrate, unless DNA analysis directly and reliably identifies a specific individual, it will always be a source of controversy in court, and the mathematical uncertainty inherent in this process is likely to be converted into errors by lawyers.
Since mathematics is an unavoidable element in trials where scientific investigations are applied, there is an urgent need to establish standards for applying mathematics.
At the same time, the public, who constitute the jury, must be properly educated to enhance their understanding of the fundamental mathematical principles essential to forensic science.
(Omitted) As some of the basic features of DNA analysis became more widely known, public understanding of DNA analysis also increased.
This demonstrates that familiarity with other elements can be similarly acquired, and the continued popularity of TV dramas related to crime investigations clearly demonstrates that people are not indifferent to this topic.
_335~336p

The mathematical errors covered here are common problems encountered not only in courtrooms but also in everyday life.
The fascinating crime cases presented in this book will help you understand the dangers and inadvertent misapplication of mathematics in everyday life.
- BBC Focus

A fascinating journey through the misuse of mathematical calculations in the courtroom.
The cases presented in this book are not only interesting in themselves, but also add mathematical understanding to solving criminal cases, doubling the fun.
From the course of events to the mathematical problems and solutions involved, and through precise analysis, you will be able to gain mathematical insight.
- Publisher's Weekly

This book offers a fascinating analysis of the frauds and forgeries involving historical figures such as Charles Ponzi, Hetty Green, and Alfred Dreyfus.
A fascinating book that illustrates how wrong human intuition can be.
- "Circus"

Schnepps and Colmez present a reconstructed criminal case based on mathematical concepts that are easy for the general public to understand, and urge the careful use of mathematics in trials.
Today, with the increasing use of sophisticated forensic techniques like DNA evidence in courts, this warning still holds true.
- Washington Independent Review of Books

It captivates many readers with its dramatic plot of a legal case involving mathematics.
A captivating book that simultaneously stimulates the mind and sparks interest in mystery.
- Mathematical Society of America