Bath Taps into Science

In 1881, Simon Newcomb noticed an unusual pattern in his book of logarithm tables, the early pages were far more worn than the latter ones.  It would be 50 years before this phenomenon would be named - not after Newcomb, as is almost traditional in mathematics, but after Frank Benford who later studied it in 1938. Although this observation may seem innocuous, Benford’s Law, as it is now known, is far-reaching and can be invaluable for detecting fraud across a range of situations.

 Benford took the initial observation even further and tested it on as many different sets of data as he could get his hands on, from the surface areas of rivers to the street addresses of the first 300 people listed in American Men of Science. Each of these conformed broadly to the same pattern.

 Put simply, Benford’s Law is the principle that numbers starting with lower digits are more common in sets of data. He theorised that in ‘real-world’ statistics the distribution of first digits would not be uniform but instead follow a trend, where numbers starting with lower digits (1, 2, 3) are more likely to arise. Using population as an example, Benford showed that in US cities the population was more likely to begin with 1. This could mean 10, or 100, or 1000 or even 1 million.

 Formally the law states that the probability of a number having a certain digit d as its first digit is given by:

 This can also be shown as a graph of percentage frequency of each of the first digits:

  This shows that numbers beginning with 1 are six times more likely than numbers beginning with 8 or 9. We can also extend this to account for the second digits in a number and this even transfers into numbers expressed in other bases, such as hexadecimal. However despite having discovered this phenomenon over a century ago, mathematicians still struggle to understand exactly why it occurs. We can see that it follows naturally from exponentially growing processes, such as the amount of bacteria in a sample at regular intervals. It also applies to data that is scale-invariant, that is to say, has the same distribution of digits even if we change our units from meters to feet, for example. Despite this, there is no reason why so many sets should have this property.

 So what has this got to do with us? One important use of this Law is in detection of fraudulent financial behaviour. It will be no surprise to you that companies and individuals within companies are often drawn to the desire to ‘cook the books’ to their own ends. However, company financial statements generally follow Benford’s Law and a comparison between these statements and the expected Benford’s distribution can therefore be used to weed out those statements which have been tampered with.

 Just like companies, countries have incentive for fudging their financial statements as well. In Greece’s case, the incentive came from their membership of the Eurozone. Members are expected to comply with certain economic benchmarks, and face sanctions if they fail to do so. In 2009, a study was launched into the economic information provided by 27 EU nations, and according to Benford’s Law, Greece’s economic data could be described as suspicious to say the least. Though this was not a surprise (the EU had launched several investigations into Greece’s economic reports already), it is an excellent example of how this Law can be used across a wide range of data sets to provide insight into their validity.

 One relatively high profile use of Benford’s Law was following the 2009 Iranian Election. Following this election, the candidate officially considered to have won was Mahmoud Ahmadinejad, however this was unexpected and resulted in outcries and riots across the world. Many have since argued that statistical analysis of the vote count can be used as evidence of tampering. In 2006, Walter Mebane of the University of Michigan studied election results from a variety of countries. When he later compared his findings to the vote count of the 2009 elections in Iran, he found that these did not match well to a Benford’s Law test on the second digits, contrary to what he had found in 2006 where most election data did comply with this distribution.

 Some 130 years later, what started as an anecdote regarding a now rarely used tool, has been used to detect fraud in financial, economic and election data on a large scale. From a nation’s economic data all the way down to your own bank statement, Benford’s Law describes so much of how our world really operates.

Modern mathematics has enabled humans to develop in very sophisticated ways. It has allowed the use of technologies to cure diseases, build complex electronic infrastructures and assist in space exploration. Whilst the crux of mathematics is embedded in the ‘number’, we still do not know how to find and prove the existence of almost every number that exists.

Philosophical statements like the above arise from confusions regarding infinity, and how there are different ways to perceive the size of something that is absurdly sizeless. Many Mathematicians of the past and the present have investigated the wonderment of infinity and have created thought-provoking experiments that still perplex people today.

One such Mathematician is David Hilbert, a German Mathematician born in 1862. Starting school two years later than most German children, at the age of 8, Hilbert did not thrive in his early years at school. He didn’t enjoy the task of memorising lots of information, which is surprising for somebody who has now become a prolific Mathematician who has developed theories across many different disciplines of Mathematics.

After entering the University of Königsberg, he soon formed a great friendship with Hermann Mikowski, a fellow student studying mathematics. Mikowski helped Hilbert with his University research, and together they supported each other’s ideas and study. After graduating and becoming a working mathematician, Hilbert spent a lot of his life researching and developing new tools in Mathematics that are still used today. In 1900 Hilbert proposed 23 different Mathematical problems (“Hilbert’s Problems”) that hadn’t been solved at the time. In the century following, all but 4 of these problems have been partly solved, each signifying great Mathematical achievement. Twenty-four years later after these problems were proposed, Hilbert gave a lecture that mentioned a thought experiment, now called Hilbert’s Paradox of the Grand Hotel.

To give some insight into his lecture, we must know what infinity means in the terms of the Grand Hotel. The counting numbers, “one, two, three, …”, also called the natural numbers that repeat forever are in a class of infinity called “countably infinite”. We could count them one after another, even if it reaches into infinity (and beyond). Any set of numbers is countably infinite if we could list them in an order and uniquely draw lines between them and the natural numbers. That is, matching each number in one set to exactly one from the other. We call this a 1-1 (“one-to-one”) correspondence between two sets, and we say that they have the same “cardinality”.

Hilbert’s idea described a popular Hotel, in which there are a countably infinite number of rooms already occupied. One guest arrives late and requests to be put in a room, but there is no empty room at the “end” to put him in. The hotel requests the guests all move into their adjacent room, allowing the new guest to check into room number 1. Later on, a coach arrives carrying a countably infinite amount of guests looking to get some sleep. They all need a room, but fitting them in seems like an impossible task, until one suggests the current guests all move to the room whose number is double their own. The new guests can then be ordered and slotted in between. Whilst a very popular destination, Hilbert’s hotel doesn’t get very good reviews as guests often complain about being moved around. If you have an eternity to read all of them, that is!

What the above seems to imply is that there is an equal amount of even numbers as there is even and odd! In Mathematical terms, we say that the cardinality of the set of all even numbers is equivalent to the cardinality of the set of all of the natural numbers. No matter how many more guests you bring in, as long as they are countably infinite, you will be able to fit them into this Grand Hotel that has every room occupied.

The other, scarier kind of infinity is that called “uncountably infinite” which cannot be listed out in any kind of order. It was first proven in 1874 by the German mathematician Georg Cantor that the real numbers, that is every number that exists in the number line you would recognise from school maths lessons, is an uncountable set. This idea was met with much controversy, as people didn’t believe that there could be anything “bigger” than the cardinality of the set of all counting numbers. David Hilbert strongly supported Georg Cantor’s ideas, recognised his genius mind and described his support with the quote

“No one shall drive us from the paradise which Cantor has created for us”

Many of the numbers that Mathematicians have been able to prove the existence of, fall within a countable set. That is, all counting numbers, all rational numbers (those that can be expressed as a ratio of two integers) and even more complicated numbers, such as the surds (for example, the square root of 2). If these all fall within a countable set, but the real number line continuum is uncountable, there is an uncountable set of numbers that contains other numbers not in one of these countable sets. One reason we cannot find these numbers easily is that they are infinitely precise, and as such to describe these numbers in decimal form would take an infinite amount of time and paper. One uncountable family we know of is called the “transcendental numbers”, where our favourite number celebrities, (pi) and (Euler’s constant to describe compound interest growth) can be found.

Transcendental numbers are those that are not “algebraic” i.e. cannot be used with algebra to solve polynomial equations with integer coefficients. (For example 2x+3=0 or x2+x-2=0). Whilst almost all numbers are transcendental, we know the existence of such a small amount of them that the numbers we do know take up no space at all in vast sea of real numbers. Trying to fit the Transcendental numbers into Hilbert’s Hotel would be an impossible task as we would have no way of ordering them, which is going to leave a lot of people unhappy and without a room for the night.

There are different number systems that we have created to either aid in real life application or more theoretical pursuits. The complex numbers were found to fill the hole left in algebra created with the roots of negative numbers. The quaternion numbers that are an extension of the complex numbers are used to assist in computer game graphic design. More abstract systems such as the hyperreal numbers and the surreal numbers extend what are common number analysis techniques into the weird and wonderful. In a world of numbers real and imaginary, knowing that we don’t know almost all of them and probably never will, hopefully leaves the wonder surrounding mathematics ever present for years and eternities to come.

When preparing for a romantic meal for two there are many things to consider. Do I have anything to wear? Where shall we eat? Who’s driving? Will we need to book a taxi? What should we talk about?

 But wait, there’s one more thing you should consider. If your inbox is anything like mine, it is inundated with discount vouchers for high street restaurants just waiting to be used. However, comparing deals for mains only, food only or a fixed amount off can quickly become confusing when you are trying to work out which one will save you the most money. So, how do you know which offers the best saving for you?

    £20 off when you spend over £50

Unlike most other deals, this takes money off the total bill and not just the food bill. This deal will be at its greatest value when spending close to £50, when there is a 40% discount. As spending increases, the discount will decrease towards 0%.

The real percentage discount is therefore relatively easy to calculate:

1/3 off Mains

This discount is only taken off the main meals ordered, so generally the overall discount will be less than 1/3 (or 33%). This deal offers the best value when most of the bill consists of main meals.

The real percentage discount here is slightly more complex to calculate:

     2 for 1 Mains

Like the voucher above, this only applies to main meals and offers the equivalent of 50% off mains (in the cases of even numbers and identically priced food). The optimal value of the discount is when most of the bill consists of main meals.

The real percentage discount here can be calculated by:

e.g.  For 4 guests the 2 cheapest meals will be deducted.

         25%/30%/40% off Food

A percentage off the food bill often represents better value for money than those offers that only apply to main meals. This usually means that only the drinks (and children’s food) will not be discounted. Therefore the larger the drinks bill, the lower the real value of the discount.

The real percentage discount can be calculated as follows:

   So, which should you use?

We’ll assume first that the vouchers are valid i.e. both of you are ordering off the correct menu, you’re in a participating restaurant, it’s the correct day of the week, and (rather unrealistically) we’ll also assume that all of these vouchers are at your disposal. We’re also assuming that both of you eat the same number of courses and they are roughly equal in cost…

Now, taking an average menu:

 Average Menu

 Starter               £5

Main  Meal        £12

Dessert              £5

Bottle  of Wine   £15

 The cost of the meal with each of the vouchers would be:

   If you both only had one course, the best deal would be 2 for 1 mains, offering 31% off the total bill. Should you opt to have two courses, then the best offer would be 40% off food, representing 28% off the total bill. Finally, for three courses the best deal would be £20 off when you spend over £50 where 34% would be taken off the final bill.

Of course, in reality you would never know in advance exactly how much you would spend on the night, but a rough idea beforehand may help you to determine the voucher offering the best value for you.

  Romantic Tip

Discount vouchers tend not to be a romantic topic to discuss on a date… so pay more attention to your partner than the percentages at the time!

Consider the following problem, originally set out by Kahneman and Tversky:

"A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are Blue and 15% are Green.

www.londoncouncils.gov.uk CC BY

 A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified colours 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?"

It might seem a simple case of multiplying the chance of the cab being blue by the chance that the witness is correct:

P(Blue)*P(Witness is correct) = 0.85*0.8 = 0.68

This answer should give you pause for thought however, surely we would expect the witness testimony to increase the chance of the cab being blue.

So, we need to look as well at the other outcome which leads to a blue car being reported:

P(Green)*P(Witness is incorrect) = 0.15*0.2 = 0.03

This means our total  probability of getting a blue report is 0.71 (71 out of 100 reports would be blue) and so our chance of a blue car being involved given that a blue report has been made is:

0.68/0.71 = 0.957 (number of these where a blue cab is actually involved out of the total reports citing a blue cab)

When Kahneman and Tversky originally asked this question they found few people correctly answered it. Further studies have found that most people instinctively equate the inverse probabilities:   P(A│B) = P(B│A),   a reaction known as "the base rate fallacy".

Conditional Probabilities & Bayes' Theorem

But what does P(A│B) actually mean? We call this the conditional probability. If we were reading it aloud we would say "the probability of A given B". This means that we are interested in the chance of an event "A" happening when we already know that an event "B" has happened. This is different to just P(A) which is the probability that A happens, in this case we do not care about any other events and whether they may or may not have happened.

Clearly in some cases the second event may have no impact on the first; for example we would expect the fact that it is raining to have no impact on getting a 6 on a dice, and likewise getting a 6 will not impact the weather. In this case there would be no difference between the conditional probability of these events and the straight probability. There are plenty of cases where this is not true however, consider flipping two coins.

The chance of us flipping two heads is 1/4 (need a head on both coins).

The chance of us having two heads if I tell you the first coin is a head is 1/2 (just need a head on the second coin).

The chance of having two heads given one of the coins is a head is actually 1/3. This may seem counter intuitive but if we list out the possible scenarios: HT, TH, HH, we see that without the information about the ordering we cannot narrow down which coin has our "known" head.

math.serpmedia.org  CC BY

 Hopefully this shows why equating the inverse conditional probabilities is not always logically sound. In order to deal with problems like our taxi problem, we need the help of Rev. Thomas Bayes, an 18th century theologian and statistician. He lends his name to a method of statistics and also  a rather useful formula. His work set out important results on conditional probability and is used in many applications today.

Importance of understanding probabilities

Along with hypothetical taxis there are several important applications of Bayes' work. It is a powerful tool in medical diagnostics; Imagine testing for a rare disease with a test that has the chance of an incorrect result.

Even if the test was correct say 99% of the time for a disease is only in 1 of 10,000 people, our chances of having the disease with a positive result are surprisingly low. In fact if we apply the theorem we would see that there is only a 1% chance of us having the disease even with a positive test.

This is because of the large number of false positives, just as with the taxi problem when we needed to consider "false" blue reports. In 10,000 people, 1 will have the disease but we will get around 100 false positives if we test everybody. So while our test might be 99% accurate, it seems fairly poor for general screening.

Ignoring Bayes' Theorem in this and equating the inverses would have had us believe that this chance is the 99% which is our probability of testing positive given we have the disease.

 To end, and to illustrate another area where this kind of thinking is important, let us return to crime and evidence.

Imagine during jury duty you are presented with this argument by the prosecution:

"We have found a sample of the murderer’s DNA, and there is a 0.1% chance of a random person’s DNA matching this sample. We have a man whose DNA does match. Therefore there is a 99.9% chance that this man is the murderer."

Is he right? And should we convict this man on this evidence alone? I shall leave you to decide.

The creator of the annual Bath Taps into Science festival, Professor Chris Budd, visited Windsor Castle last week to collect an OBE for services to science and maths education.

The Queen awarded Professor Budd his honour, at a ceremony that lasted around an hour. He was accompanied on the day by his wife, daughter and mother.

Bath Taps into Science

Professor Budd is one of the founders of Bath Taps into Science, a major hands-on science festival which has won several national prizes in the 14 years it has been running.

The week of events aims to show students and families how the science and maths that they learn at school can be applied to the wider world and to inspire them to want to become a ‘scientist’ or ‘mathematician’.

In 2010, Professor Budd led the hands on maths exhibition 'Living in a Complex World' which was part of the Royal Society’s 350th anniversary celebration, attended by 70,000 members of the public. Following its success, the team has since taken it to Manchester, Japan, Canada, Ireland and (every year from 2011) to the Big Bang Fair, at which it has been presented to 300,000 members of the general public.

Public engagement

As a Higher Education Academy National Teaching Fellow he set up a Communicating Maths undergraduate course and, as the Royal Institution Professor of Maths since 2000, has spoken at maths masterclasses around the country. He recruits students from the UK maths community to give talks to the general public and works tirelessly for public engagement to be regarded as important as other aspects of academia.  In 2013 he was the mathematical consultant for the 150th anniversary of the London Underground, designing a maths trail for the underground network.

Professor Budd OBE collected his honour from the Queen at Windsor Castle. (Left to right: Professor Budd’s wife Sue, Professor Chris Budd, his daughter Bryony, and his mother Jillian)

He is very active in both the Institution of Mathematics and its applications (initiating and co-directing the first UK Festival of Maths in July 2014 and as Vice-President coordinating a year-long celebratory event for its 50th anniversary); and the London Mathematical Society for which he was Education Secretary. He was also part of the small team which produced the well-regarded Vorderman report on the state of maths in schools commissioned by Michael Gove.

From meteorology to WiFi

Professor Budd has published over 100 research papers, raised over £5m in research funding and founded (and still runs) the University’s MSc in Modern Applied Mathematics. His work has always been centred on making mathematics directly relevant to the needs of society, and he collaborates closely with many different industries and bodies to do this, working on problems as varied as microwave cooking, cancer treatment, land mine detection and how to make WiFi more reliable.

He is currently working on solving problems in meteorology and climate change, and his close collaboration with the Met Office has helped to improve weather forecasting accuracy. He co-founded an international network which brings together mathematicians working in climate change with policy-makers (and leads its outreach programme). He is also the UK lead on a Marie Curie network, which trains many PhD students across Europe.

"An amazing day"

Professor Budd said: “It was an amazing day - I met lots of incredible people who’d done wonderful things.

“I was very honoured to meet the Queen herself and she seemed very interested in the Bath Taps into Science fair.”

Commenting on his work in communicating mathematics to the public, Professor Budd said: “I try to show people that maths is all around them and relevant to the real world.

“Teachers should be given the opportunity to explore the creative side of mathematics, rather than teaching it in the traditional rather formulaic way.

“Maths is increasingly important in our lives and will become more so with the big data revolution. The University of Bath is engaging strongly with this, with next week’s launch of the Bath Institute for Mathematical Innovation which will put mathematicians in touch with industry to work on real life problems.”