Category: Skills and Learning

From avoiding the number seven to picking numbers over 31, mathematician Peter Rowlett has a few psychological strategies for improving your chances when playing the lottery.

Would you think I was daft if I bought a lottery ticket for the numbers 1, 2, 3, 4, 5 and 6? There is no way those are going to be drawn, right? That feeling should – and, mathematically, does – actually apply to any set of six numbers you could pick.

Lotteries are ancient. Emperor Augustus, for example, organised one to fund repairs to Rome. Early lotteries involved selling tickets and drawing lots, but the idea of people guessing which numbers would be drawn from a machine comes from Renaissance Genoa. A common format is a game that draws six balls from 49, studied by mathematician Leonhard Euler in the 18th century.

The probabilities Euler investigated are found by counting the number of possible draws. There are 49 balls that could be drawn first. For each of these, there are 48 balls that can be drawn next, so there are 49×48 ways to draw two balls. This continues, so there are 49×48×47×46×45×44 ways to draw six balls. But this number counts all the different arrangements of any six balls as a unique solution.

How many ways can we rearrange six balls? Well, we have six choices for which to put first, then for each of these, five choices for which to put second, and so on. So the number of ways of arranging six balls is 6×5×4×3×2×1, a number called 6! (six factorial). We divide 49×48×47×46×45×44 by 6! to get 13,983,816, so the odds of a win are near 1 in 14 million.

Since all combinations of numbers are equally likely, how can you maximise your winnings? Here is where maths meets psychology: you win more if fewer people share the prize, so choose numbers others don’t. Because people often use dates, numbers over 31 are chosen less often, as well as “unlucky” numbers like 13. A lot of people think of 7 as their favourite number, so perhaps avoid it. People tend to avoid patterns so are less likely to pick consecutive or regularly spaced numbers as they feel less random.

In July, David Cushing and David Stewart at the University of Manchester, UK, published a list of 27 lottery tickets that guarantee a win in the UK National Lottery, which uses 59 balls and offers a prize for matching two or more. But a win doesn’t always mean a profit – for almost 99 per cent of possible draws, their tickets match at most three balls, earning prizes that may not exceed the cost of the tickets!

So, is a lottery worth playing? Since less than half the proceeds are given out in prizes, you would probably be better off saving your weekly ticket money. But a lecturer of mine made an interesting cost-benefit argument. He was paid enough that he could lose the cost of a ticket each week without really noticing. But if he won the jackpot, his life would be changed. So, given that lottery profit is often used to support charitable causes, it might just be worth splurging.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett*

If you want your cup of tea to stay as hot as possible, should you put milk in immediately, or wait until you are ready to drink it? Katie Steckles does the sums.

Picture the scene: you are making a cup of tea for a friend who is on their way and won’t be arriving for a little while. But – disaster – you have already poured hot water onto a teabag! The question is, if you don’t want their tea to be too cold when they come to drink it, do you add the cold milk straight away or wait until your friend arrives?

Luckily, maths has the answer. When a hot object like a cup of tea is exposed to cooler air, it will cool down by losing heat. This is the kind of situation we can describe using a mathematical model – in this case, one that represents cooling. The rate at which heat is lost depends on many factors, but since most have only a small effect, for simplicity we can base our model on the difference in temperature between the cup of tea and the cool air around it.

A bigger difference between these temperatures results in a much faster rate of cooling. So, as the tea and the surrounding air approach the same temperature, the heat transfer between them, and therefore cooling of the tea, slows down. This means that the crucial factor in this situation is the starting condition. In other words, the initial temperature of the tea relative to the temperature of the room will determine exactly how the cooling plays out.

When you put cold milk into the hot tea, it will also cause a drop in temperature. Your instinct might be to hold off putting milk into the tea, because that will cool it down and you want it to stay as hot as possible until your friend comes to drink it. But does this fit with the model?

Let’s say your tea starts off at around 80°C (176°F): if you put milk in straight away, the tea will drop to around 60°C (140°F), which is closer in temperature to the surrounding air. This means the rate of cooling will be much slower for the milky tea when compared with a cup of non-milky tea, which would have continued to lose heat at a faster rate. In either situation, the graph (pictured above) will show exponential decay, but adding milk at different times will lead to differences in the steepness of the curve.

Once your friend arrives, if you didn’t put milk in initially, their tea may well have cooled to about 55°C (131°F) – and now adding milk will cause another temperature drop, to around 45°C (113°F). By contrast, the tea that had milk put in straight away will have cooled much more slowly and will generally be hotter than if the milk had been added at a later stage.

Mathematicians use their knowledge of the rate at which objects cool to study the heat from stars, planets and even the human body, and there are further applications of this in chemistry, geology and architecture. But the same mathematical principles apply to them as to a cup of tea cooling on your table. Listening to the model will mean your friend’s tea stays as hot as possible.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Katie Steckles*

Have you ever struggled with teaching statistics? Do you and your students share a sense of apprehension when data lessons appear in the scheme of work? You’re not alone. Anecdotally, many teachers tell me that statistics is one of the topics they like teaching the least, and I am no exception to this myself. In my mathematics degree, I took the minimum number of statistics-related courses allowed, following a very poor diet of data at school, and carried this negative association into my teaching. Looking back on my career in the classroom, I did not do a good job of teaching statistics, but having had the luxury of spending many years at Cambridge Mathematics immersed in research from excellent statistics teachers and education academics I now understand why!

So now, of course, the question has been posed. Why is statistics hard to teach well? In part, I believe that it stems from viewing statistics through a mathematical lens – understandably, given that we are delivering it alongside quadratic equations, Pythagoras’ theorem, fractions, decimals and percentages. But while statistical analysis would not exist without the mathematical concepts and techniques underpinning it, we have a tendency within curricula to make the mathematical techniques the whole point, and reduce the statistical analysis part to an afterthought or an added extra. Students find the more subjective analysis hard, so it is tempting to make sure everyone can manage the techniques and then focus on the interpretation as something only the most able have time to spend on (although, there is always the additional temptation to move on to other, more properly ‘maths-y’ topics as soon as possible).

This approach is at odds with how education researchers suggest students should encounter statistical ideas. In the early 1990s, George Cobbⁱ and other researchers recommended that statistics should

• emphasise statistical thinking,
• include more real data,
• encourage the exploration of genuine statistical problems, and
• reduce emphasis on calculations and techniques.

Since then, much subsequent research has refined these recommendations to account for new technology tools and new ideas, but the core principles have remained the same. In much of my reading of education research, three ways of seeing or interacting with data keep appearing:

·        Data modelling – the idea that data can be used to create models of the world in order to pose and answer questions

·        Informal inference – the idea that data can be used to make predictions about something outside of the data itself with some attempt made to describe how likely the prediction is to be true

·        Exploratory data analysis – the idea that data can be explored, manipulated and represented to identify and make visible patterns and associations that can be interpreted

In the abstract, these ways of seeing, while distinct, have a degree of overlap, and all students may benefit from multiple experiences of all three approaches to data work from their very earliest encounters with data through to advanced-level study.

Imagine the following classroom activity that could be given to very young students (e.g., in primary school). A class of students is given a list of snacks and treats and the students are asked to rank them on a scale of one to five based on how much they like each item. How could this data be worked with through each of the three approaches?

Firstly, we will consider data modelling. Students could be asked to plan a class party with a limited budget. They can buy some but not all of the items listed and must decide what they should buy so that the maximum number of students get to have things they like. In this activity, students must create a model from the data that identifies those things they should buy more of, and those things they should buy least of, along with how many of each thing they should get – perhaps considering these quantities proportionally. This activity uses the data as a model but inevitably requires some assumptions and the creation of some principles. Is the goal to ensure everyone gets the thing they like most? Or is it to minimise the inclusion of the things students like least? What if everyone gets their favourite thing except one student who gets nothing they like?

Secondly, we will think about this as an activity in informal inference. Imagine a new student is joining the class and the class wants to make a welcome pack of a few treats for this student, but they don’t know which treats the student likes. Can they use the data to decide which five items an unknown student is most likely to choose? What if they know some small details about the student; would that additional information allow them to decide based on ‘similar’ students in the class? While the second part of this activity must be handled with a degree of sensitivity, it is an excellent primer for how purchasing algorithms, which are common in online shops, work.

Finally, we turn to exploratory data analysis. In this approach students are encouraged to look for patterns in the data, perhaps by creating representations. This approach may come from asking questions – e.g., do students who like one type of chocolate snacks rate the other chocolate snacks highly too? Is a certain brand of snack popular with everyone in the class? What is the least popular snack? Alternatively, the analysis may generate questions from patterns that are spotted – e.g. why do students seem to rate a certain snack highly? What are the common characteristics of the three most popular snacks?

Each of these approaches could be engaged in as separate and isolated activities, but there is also the scope to combine them and use the results of one approach to inform another. For example, exploratory data analysis may usefully contribute both to model building and inference making and support students’ justifications for their decisions in those activities. Similarly, data modelling activities can be extended into inferential tasks very easily, simply by shifting the use of the model from the population of the data (e.g., the students in the class it was collected from) to some secondary population (e.g., another class in the school, or as in the example, a new student joining the class).

Looking back on my time in the classroom, I wish that my understanding of these approaches and their importance for developing statistical reasoning skills in my students had been better. While not made explicit as important in many curricula, there are ample opportunities to embed these approaches and make them a fundamental part of the statistics teacher’s pedagogy.

Do you currently use any of these approaches in your lessons? Can you see where you might use them in the future? And how might you adapt activities to allow your students opportunities to engage in data modelling, informal inference and exploratory data analysis?

 For more insights like this, visit our website at www.international-maths-challenge.com.

Credit for the article given to Darren Macey

Artificial Intelligence (AI) is becoming more and more prevalent in our daily lives, and there are many AI-powered websites that can help school students improve their math skills. Here are the best AI sites for school students to help them develop their maths skills

Khan Academy (https://www.khanacademy.org/)

Khan Academy is a non-profit educational website that offers a wide range of math courses and resources for students of all ages. Their math courses are designed to be interactive, engaging, and accessible, making it easy for students to learn at their own pace. Khan Academy uses AI-powered algorithms to provide personalized recommendations for each student, ensuring that they are working on the concepts that they need to improve.

DreamBox Learning (https://www.dreambox.com/)

DreamBox Learning is an AI-powered math education platform that uses adaptive learning to provide personalized math lessons to students. The platform uses AI algorithms to analyze a student’s performance and provide personalized feedback and recommendations, ensuring that they are working on the concepts they need to improve. DreamBox Learning is designed for students from kindergarten through 8th grade.

IXL Math (https://www.ixl.com/math/)

IXL Math is an AI-powered math education platform that offers a wide range of math courses and resources for students of all ages. The platform uses AI algorithms to analyze a student’s performance and provide personalized recommendations for each student. IXL Math is designed to be interactive, engaging, and accessible, making it easy for students to learn at their own pace.

Matific (https://www.matific.com/)

Matific is an AI-powered math education platform that uses gamification to make math learning fun and engaging for students. The platform uses AI algorithms to analyze a student’s performance and provide personalized feedback and recommendations, ensuring that they are working on the concepts they need to improve. Matific is designed for students from kindergarten through 6th grade.

Prodigy (https://www.prodigygame.com/)

Prodigy is an AI-powered math game that helps students learn math in a fun and engaging way. The game uses AI algorithms to analyze a student’s performance and provide personalized recommendations for each student. Prodigy is designed for students from 1st through 8th grade and covers a wide range of math concepts.

There are many AI-powered websites that can help school students improve their math skills. These platforms use AI algorithms to provide personalized recommendations, feedback, and resources, ensuring that each student is working on the concepts they need to improve. By using these websites, students can improve their math skills and develop a love for learning that will serve them well throughout their academic careers.

Relating mathematics to questions that are relevant to many students can help address its image problem, argues Eugenia Cheng.

Mathematics has an image problem: far too many people are put off it and conclude that the subject just isn’t for them. There are many issues, including the curriculum, standardised tests and constraints placed on teachers. But one of the biggest problems is how maths is presented, as cold and dry.

Attempts at “real-life” applications are often detached from our daily lives, such as arithmetic problems involving a ludicrous number of watermelons or using a differential equation to calculate how long a hypothetical cup of coffee will take to cool.

I have a different approach, which is to relate abstract maths to questions of politics and social justice. I have taught fairly maths-phobic art students in this way for the past seven years and have seen their attitudes transformed. They now believe maths is relevant to them and can genuinely help them in their everyday lives.

At a basic level, maths is founded on logic, so when I am teaching the principles of logic, I use examples from current events rather than the old-fashioned, detached type of problem. Instead of studying the logic of a statement like “all dogs have four legs”, I might discuss the (also erroneous) statement “all immigrants are illegal”.

But I do this with specific mathematical structures, too. For example, I teach a type of structure called an ordered set, which is a set of objects subject to an order relation such as “is less than”. We then study functions that map members of one ordered set to members of another, and ask which functions are “order-preserving”. A typical example might be the function that takes an ordinary number and maps it to the number obtained from multiplying by 2. We would then say that if x < y then also 2x < 2y, so the function is order-preserving. By contrast the function that squares numbers isn’t order-preserving because, for example, -2 < -1, but (-2)² > (-1)². If we work through those squaring operations, we get 4 and 1.

However, rather than sticking to this type of dry mathematical example, I introduce ones about issues like privilege and wealth. If we think of one ordered set with people ordered by privilege, we can make a function to another set where the people are now ordered by wealth instead. What does it mean for that to be order-preserving, and do we expect it to be so? Which is to say, if someone is more privileged than someone else, are they automatically more wealthy? We can also ask about hours worked and income: if someone works more hours, do they necessarily earn more? The answer there is clearly no, but then we go on to discuss whether we think this function should be order-preserving or not, and why.

My approach is contentious because, traditionally, maths is supposed to be neutral and apolitical. I have been criticised by people who think my approach will be off-putting to those who don’t care about social justice; however, the dry approach is off-putting to those who do care about social justice. In fact, I believe that all academic disciplines should address our most important issues in whatever way they can. Abstract maths is about making rigorous logical arguments, which is relevant to everything. I don’t demand that students agree with me about politics, but I do ask that they construct rigorous arguments to back up their thoughts and develop the crucial ability to analyse the logic of people they disagree with.

Maths isn’t just about numbers and equations, it is about studying different logical systems in which different arguments are valid. We can apply it to balls rolling down different hills, but we can also apply it to pressing social issues. I think we should do both, for the sake of society and to be more inclusive towards different types of student in maths education.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Eugenia Cheng*

Mathematics is an essential part of our daily lives, and it is crucial for children to develop strong math skills from a young age. Here are some reasons why math is important in kids’ daily lives:

Problem-solving skills: Mathematics teaches children how to solve problems, both in math-related situations and in real-life situations. The logical and analytical skills they develop through math help them find solutions to problems and make informed decisions.

Money management: Math skills are essential for managing finances. Children need to learn how to add, subtract, multiply, and divide money to manage their allowances and understand the value of different amounts.

Time management: Math skills also play a critical role in time management. Children need to be able to tell time, calculate elapsed time, and understand the concept of time zones to manage their schedules and keep appointments.

Measurements: Measurements are everywhere, from cooking to construction. Math skills are necessary for children to understand the different units of measurement and use them in everyday situations.

Technology: Math is essential for understanding and using technology. Programming, robotics, and computer science are all based on math concepts. Register for the International Maths Olympiad Challenge to improve your kid’s skill and thinking level.

Academic and career success: Strong math skills are essential for success in academic and career fields such as engineering, science, finance, and technology. Building a strong foundation in math from a young age can set children up for future success.

In summary, math is an essential subject that plays a crucial role in students’ academic and personal development. It helps students develop problem-solving and critical thinking skills, enhances their quantitative abilities, improves decision-making abilities, advances career opportunities, and improves overall academic performance.

One week after DeepMind revealed an algorithm for multiplying numbers more efficiently, researchers have an even better way to carry out the task.

Multiplying numbers is a common computational problem

A pair of researchers have found a more efficient way to multiply grids of numbers, beating a record set just a week ago by the artificial intelligence firm DeepMind.

The company revealed on 5 October that its AI software had beaten a record that had stood for more than 50 years for the matrix multiplication problem – a common operation in all sorts of software where grids of numbers are multiplied by each other. DeepMind’s paper revealed a new method for multiplying two five-by-five matrices in just 96 multiplications, two fewer than the previous record.

Jakob Moosbauer and Manuel Kauers at Johannes Kepler University Linz in Austria were already working on a new approach to the problem prior to that announcement.

Their approach involves running potential multiplication algorithms through a process where multiple steps in the algorithm are tested to see if they can be combined.

“What we do is, we take an existing algorithm and apply a sequence of transformations that at some point can lead to an improvement. Our technique works for any known algorithm, and if we are lucky, then [the results] need one multiplication less than before,” says Moosbauer.

After DeepMind published its breakthrough, Moosbauer and Kauers used their approach to improve on DeepMind’s method, slicing off another step to set a new record of 95 multiplications. They have published the proof in a pre-print paper, but haven’t yet released details of the approach they used to find improvements on previous methods.

“We wanted to publish now to be the first one out there, because if we can find it in such a short amount of time, there’s quite some risk that we get outdone by someone else again,” says Moosbauer.

The latest paper is entirely focused on five-by-five matrix multiplication, but the method is expected to bring results for other sizes. The researchers say that they will publish details of their technique soon.

Moosbauer says that DeepMind’s approach brought fresh impetus to an area of mathematics that hadn’t been receiving much attention. He hopes that other teams are also now working in a similar vein.

Matrix multiplication is a fundamental computing task used in virtually all software to some extent, but particularly in graphics, AI and scientific simulations. Even a small improvement in the efficiency of these algorithms could bring large performance gains, or significant energy savings.

DeepMind claimed to have seen a boost in computation speed of between 10 and 20 per cent on certain hardware such as an Nvidia V100 graphics processing unit and a Google tensor processing unit v2. But it said that there was no guarantee that similar gains would be seen on everyday tasks on common hardware. Moosbauer says he is sceptical about gains in common software, but that for large and specialised research tasks there could be an improvement.

DeepMind declined a request for an interview about the latest paper, but its researcher Alhussein Fawzi said in a statement: “We’ve been overwhelmed by the incredible reaction to the paper. Our hope was that this work would open up the field of algorithmic discovery to new ideas and approaches. It’s fantastic to see others exploring ideas in this space as well as building on our work so quickly.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Matthew Sparkes*

Credit: Can you find the center of a shape? You’ll be able to–even for the oddest oblong creation–with this simple science activity. No strings attached (okay, maybe one)!  George Retseck

A centering science activity

Introduction
With a little time, you can probably find the center of simple shapes such as circles and squares pretty easily. But how do you find the “middle” of an irregular shape such as a drawing of a dog or a cat? This project will show you how to do it using nothing but string and paper clips!

Background
How do you define the exact center of an object? One way to do this is to find the object’s center of mass. The center of mass is the point about which an object will balance if you try to rest it on your fingertip. Or if you hang an object, for example a picture frame from a nail, the center of mass will hang directly below the nail.

For symmetrical objects, finding the center of mass is relatively easy. For example, for a rectangular picture frame, you know the center of mass is in the middle of the rectangle and you can find that with a ruler. When you hang the picture frame, you will make sure it is centered on the nail—otherwise it will tip to one side and will be off-center. The same applies to other symmetrical objects such as a spherical basketball; you know the center of mass is in the middle of the sphere.

What about irregularly shaped objects such as a dog or cat or person? Now finding the center of mass is not so easy! This activity will show you how to find the center of mass for any two-dimensional shape you cut out of paper using a trick that has to do with the hanging picture frame mentioned above. If you hang a shape from a single point, you know the center of mass will always rest directly below that point. So, if you hang a shape from two different points (one at a time) and draw a line straight down from each point, the center of mass is where those lines intersect. This technique can be used for any irregular two-dimensional shape. Don’t believe it? Try this activity to find out!

Materials

• Paper (Heavier paper, such as construction paper, card stock or thin cardboard from the side of a cereal box will work best.)
• Scissors (Have an adult help with cutting if necessary—especially on thicker materials.)
• String
• Pencil
• Ruler
• Two paper clips or a pushpin and another small, relatively heavy object you can tie to the string (such as a metal washer)

Preparation

• Cut a piece of string about one foot long and tie a paper clip to each end. (Alternatively, you can use any other small object such as a metal washer on one end—this will serve as a weight—and any other small, pointy object like a needle or pushpin on the other end—this will be used to puncture the paper.)

Procedure

• Start with an easy shape: Cut out a rectangular piece of paper or cardboard. Can you guess where the center of mass of the rectangle is? If so, use a ruler to measure where you think it will be and mark this spot with your pencil.
• Punch several small holes around the edge of the paper. Make them as close to the edge as possible without ripping the paper. (This is important for the accuracy of this technique). The exact location of the holes does not matter but this technique will work best if you space them all the way around the edge (not just put two holes right next to each other).
• Now poke one end of one paper clip (or pushpin) through one of the holes to act like a hanging hook. Make sure the paper can swing easily from the hook and does not get stuck (Rotate it back and forth a few times to loosen the hole if necessary).
• Hold on to your “hook” and hold the paper up against the wall. Let the paper swing freely and make sure the string can hang straight down and does not get stuck.
• Use a pencil and ruler to draw a straight line on the paper along the string. Does this line go through the center of mass you predicted earlier?
• Now, hang your paper from a different hole and repeat the process. Where does this line intersect the first line?
• Repeat the process several more times with different holes. Do all the lines intersect at the same point?
• Now cut out an irregular shape. You can cut out a “blob” or draw something like a dog or cat and then cut out the outline. Make sure the shape you cut out remains stiff and flat. (That is, do not cut very thin sections that might be floppy.) Can you use a ruler to predict where the center of mass of your irregular shape will be? This is much harder!
• Punch holes around the edge of your irregular shape and repeat the activity. One at a time, hang the shape and the string from one of the holes and draw a line along the string. Where do the lines intersect? Does this match up with what you predicted?
• Extra: If you use a stiff enough material to cut out your shape (such as cardboard), can you try balancing it on your fingertip at the center of mass? What happens if you try to balance it about another point?

Observations and results
You should have found that the center of mass of the rectangle is right in the middle of the piece—halfway along the width and halfway along the height. You can easily locate this spot with a ruler. Then, when you hang the rectangle from a hole on its edge, the string should always pass through this point, regardless of which hole you use. Whereas it is much harder to predict the center of mass for an irregular shape, the same principle holds true.

Regardless of what point you hang the irregular shape from, the string will always pass through the center of mass. So, if you hang it from two or more points (one at a time), you can find the intersection of these lines—and that is the center of mass.

Note that due to small variables in the activity (such as friction on the hook that prevents the paper from rotating perfectly or the holes not being close enough to the edge of the paper), if you draw multiple lines, they might not all intersect in exactly the same place but they should still be fairly close to one another.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Ben Finio & Science Buddies

The probabilistic revolution first kicked off in the 1600s, when gamblers realized that estimating the likelihood of an event could give them an edge in games of chance.

Today, statistics has become the dominant way to communicate scientific findings. But courts can be hesitant to incorporate statistical evidence into decisions. Indeed, they have historically been antagonistic toward probabilities and are loath to be swindled by slippery statistics.

However, as an educator of statistics who has consulted in a variety of contexts and has served as expert witness to the U.S. District Court in Montana, I find that both my experience and my review of the evidence suggest that courts increasingly feature statistical thinking – whether or not it is identified as such.

Society needs to prioritize educating juries in the language of statistics. Otherwise, juries will be forever at the mercy of convincing, yet potentially invalid, testimony. Courtroom decisions should be based on facts and probabilities, not manipulation by a skilled prosecutor or defense attorney.

Thinking statistically

Probabilities changed the way human beings thought about outcomes. They are a useful tool for expressing our uncertainty about events in the world.

Will it rain today? It will or it will not, that much is certain. But probability allows us to express our ignorance about whether it will rain and quantify the degree to which we are uncertain. Stating “it will probably rain today” constituted a very innovative and different way of thinking.

Probabilities play a role in our daily lives, in decisions from whether to take an umbrella to work to whether to purchase flood insurance. We can consider “statistical thinking” to be any situation where probabilities are involved.

To some extent, humans are intuitive statisticians. For instance, research suggests we can revise a belief in the light of new evidence as prescribed by a statistical theorem, if the probabilities are given in a relatively intuitive rather than abstract fashion.

Statistical reasoning pervades many of the conclusions we draw regarding scientific phenomena. Even physics has had to acknowledge the reality of probabilities. So, if the courts use scientific findings as evidence, probabilities should naturally make their way into courtroom decisions.

Evaluating the evidence

If juries do not understand the nature of statistical conclusions, then they will be tempted to believe that scientific evidence is conclusive and deterministic, rather than probabilistic. For example, probabilities show us that cigarette smoking does not necessarily lead to cancer. Rather, extensive nicotine addiction likely leads to cancer.

Heads or tails? armydre2008/flickr, CC BY

Evidence can only fit a theory probabilistically. If we flip a coin 10 times and get 10 heads in a row, that suggests the coin may not be fair, but does not “prove” that it is biased.

Consider the analysis of DNA found at the crime scene. Is the DNA that of the accused? Maybe. Not definitively. A statistician might say, “The probability of this degree of DNA match occurring by chance is extremely small. The match may be due to chance, but since this probability is so small, we may conclude that it likely did not occur by chance, and use it as evidence against the accused.”

Of course, human judgment is fickle. Until jurors are trained to make rational decisions based on facts and probabilities, they will continue to be easily swayed by convincing litigators.

In the 1995 trial of OJ Simpson, for example, the bloody gloves found at the crime scene constituted powerful evidence against the accused. The samples obtained were extremely likely to belong to the defendant.

A statistically educated jury would not fall for Johnnie Cochran’s classic defense: “If it does not fit, you must acquit.” They would know in advance that no evidence, whatever the kind, fits a theory perfectly.

Cochran’s statement was, statistically speaking, utter nonsense. Of course no model fits perfectly, but which is the more probable model? That’s the task jurors ultimately face, even if they often perceive it as a “guilt” versus “no guilt” decision.

Whenever courts work with DNA matches, they must incorporate acceptable risk and error. But if such uncertainty can be quantified accurately, then it can serve as an aid in decision-making.

Statistical thinking indeed plays a role in the decision between guilt and innocence in a criminal trial. When a jury renders a “guilty” verdict, there is always the chance that the accused is not guilty, but that the many circumstances of the case simply lined up against him or her to lead the jury to a guilty verdict. In other words, the probability of the observed evidence under the assumption of innocence is so low that the evidence likely occurred under a more probable “narrative” – that of guilt.

But, when we make such a decision, we do so with a risk of error. This could be quite devastating to a defendant falsely put to death when all along he or she was innocent. For example, when researchers applied DNA testing to death row inmates in Illinois, they found that the results exonerated several inmates.

Errors in probability-based decisions can indeed be costly. Without a grasp of how virtually all decisions are based on probabilistic thinking, no jury can be expected to adequately assess any evidence in a rational way.

Base rates

Courts also struggle with whether and how to use base rates, another type of statistical tool.

A base rate is the probability of some characteristic being present in the population. For instance, say an individual takes a diagnostic test for a disease, such as HIV. The probability that she has the disease would be higher if she were sampled from a high-risk group – for example, if she shares needles to support a drug addiction, or engages in promiscuous sex with risky partners.

Courts often ignore base rate information. In Stephens v. State in 1989, the Wyoming Supreme Court heard testimony that “80 to 85 percent of child sexual abuse is committed by a close relative of the child.” They ultimately dismissed this, concluding that it was difficult to understand how statistical information would help reach a decision in an individual case.

In another case, a justice of the Minnesota Supreme Court proclaimed that she was “at a loss to understand” how base rates could help predict whether a particular person posed a danger to the public.

Part of the problem is that this information can appear biased against the accused. For instance, consider again the defendant accused of child sexual abuse. The probability that he is guilty might be evaluated in light of the fact that most perpetrators of abuse are relatives of or closely related to the family. This could be interpreted as biasing the evaluation against the accused. However, the courts have considered base rates in employment discrimination cases, an area where perhaps this information seems more naturally relevant (for example, Hazelwood School District v. United States).

If the courts are willing to use base rate information in discrimination cases, they should be encouraged to consider them in other cases as well, even if they seem less intuitive.

Learning to think statistically

Courts should make it a priority to instruct juries on how to interpret probabilistic evidence, so that they are not at the mercy of a convincing, yet potentially misleading, prosecutor or expert witness.

For example, juries might learn elementary statistics through coin-flipping lessons. This could help them, at minimum, find a way to think about the usual “beyond a reasonable doubt” instruction in a criminal trial.

When the assumption of innocence is rejected in favor of guilt, one does so with a risk of being wrong. How much risk is a jury willing to tolerate? Five percent? One percent? Surely such risk must also depend on the severity of the proposed punishment. Every decision is an exercise in risk and cost benefit analysis.

Until juries learn elements of statistical thinking, they are likely to continue making verdict decisions without the appropriate framework in mind. Probabilities have taken over the world, and this fact needs to be recognized by the courts.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Daniel J. Denis

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Prime numbers are more than just numbers that can only be divided by themselves and one. They are a mathematical mystery, the secrets of which mathematicians have been trying to uncover ever since Euclid proved that they have no end.

An ongoing project – the Great Internet Mersenne Prime Search – which aims to discover more and more primes of a particularly rare kind, has recently resulted in the discovery of the largest prime number known to date. Stretching to 23,249,425 digits, it is so large that it would easily fill 9,000 book pages. By comparison, the number of atoms in the entire observable universe is estimated to have no more than 100 digits.

The number, simply written as 2⁷⁷²³²⁹¹⁷-1 (two to the power of 77,232,917, minus one) was found by a volunteer who had dedicated 14 years of computing time to the endeavour.

You may be wondering, if the number stretches to more than 23m digits, why we need to know about it? Surely the most important numbers are the ones that we can use to quantify our world? That’s not the case. We need to know about the properties of different numbers so that we can not only keep developing the technology we rely on, but also keep it secure.

Secrecy with prime numbers

One of the most widely used applications of prime numbers in computing is the RSA encryption system. In 1978, Ron Rivest, Adi Shamir and Leonard Adleman combined some simple, known facts about numbers to create RSA. The system they developed allows for the secure transmission of information – such as credit card numbers – online.

The first ingredient required for the algorithm are two large prime numbers. The larger the numbers, the safer the encryption. The counting numbers one, two, three, four, and so on – also called the natural numbers – are, obviously, extremely useful here. But the prime numbers are the building blocks of all natural numbers and so even more important.

Take the number 70 for example. Division shows that it is the product of two and 35. Further, 35 is the product of five and seven. So 70 is the product of three smaller numbers: two, five, and seven. This is the end of the road for 70, since none of these can be further broken down. We have found the primal components that make up 70, giving its prime factorisation.

Multiplying two numbers, even if very large, is perhaps tedious but a straightforward task. Finding prime factorisation, on the other hand, is extremely hard, and that is precisely what the RSA system takes advantage of.

Suppose that Alice and Bob wish to communicate secretly over the internet. They require an encryption system. If they first meet in person, they can devise a method for encryption and decryption that only they will know, but if the initial communication is online, they need to first openly communicate the encryption system itself – a risky business.

However, if Alice chooses two large prime numbers, computes their product, and communicates this openly, finding out what her original prime numbers were will be a very difficult task, as only she knows the factors.

So Alice communicates her product to Bob, keeping her factors secret. Bob uses the product to encrypt his message to Alice, which can only be decrypted using the factors that she knows. If Eve is eavesdropping, she cannot decipher Bob’s message unless she acquires Alice’s factors, which were never communicated. If Eve tries to break the product down into its prime factors – even using the fastest supercomputer – no known algorithm exists that can accomplish that before the sun will explode.

The primal quest

Large prime numbers are used prominently in other cryptosystems too. The faster computers get, the larger the numbers they can crack. For modern applications, prime numbers measuring hundreds of digits suffice. These numbers are minuscule in comparison to the giant recently discovered. In fact, the new prime is so large that – at present – no conceivable technological advancement in computing speed could lead to a need to use it for cryptographic safety. It is even likely that the risks posed by the looming quantum computers wouldn’t need such monster numbers to be made safe.

It is neither safer cryptosystems nor improving computers that drove the latest Mersenne discovery, however. It is mathematicians’ need to uncover the jewels inside the chest labelled “prime numbers” that fuels the ongoing quest. This is a primal desire that starts with counting one, two, three, and drives us to the frontiers of research. The fact that online commerce has been revolutionised is almost an accident.

The celebrated British mathematician Godfrey Harold Hardy said: “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics”. Whether or not huge prime numbers, such as the 50th known Mersenne prime with its millions of digits, will ever be found useful is, at least to Hardy, an irrelevant question. The merit of knowing these numbers lies in quenching the human race’s intellectual thirst that started with Euclid’s proof of the infinitude of primes and still goes on today.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Ittay Weiss