The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most *Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world,
find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

## Statistics and probability theory

This teacher package brings together all *Plus* articles on statistics and probability theory. In addition to the *Plus* articles, the *try it yourself* section provides links to related problems on our sister site NRICH.

We've grouped the articles into three categories:

- Fun and games — The articles in this category use stats and probability to understand games of chance, sports, and strange coincidences;
- Understanding life — This category explores how stats and probability are used to understand all aspects of life, from death and disease to fraud.
- Lies, damn lies — This category focuses on those pitfalls and conundrums that sometimes contribute to our mistrust of stats.

We also present a list of relevant problems from our sister site NRICH.

### Fun and games

Winning odds — When you flip a coin we assume it has equal chance of coming up head or tails, so any coin flipping game should be a fair one. This article explores what can give you the winning advantage.

Bluffing and exploitation: An introduction to poker maths — If you've never played poker, you probably think that it's a game for degenerate gamblers and cigar-chomping hustlers in cowboy hats. But it turns out that poker is actually a very complicated game indeed.

Curious dice — This article looks at a set of unusual dice and a two-player game in which you will always have the advantage.

Struggling with chance — A 1 in 14 million chance to win the lottery, a 5% risk of cancer, a 50:50 chance of heads on a coin — we deal with probabilities all the time, but do they actually mean anything? We explore the philosophy of probability in this pair of articles, Struggling with chance and Still struggling with chance.

The logic of drug testing — Doping is a major issue in sport and drug tests are being used to find the culprits. But when an athlete fails a test can we really conclude that they are cheating?

Understanding uncertainty: how psychic was Paul? — England's performance in the 2010 World Cup was thankfully overshadowed by the attention given to Paul the octopus, who was reported as making an unbroken series of correct predictions of match winners. This article looks at Paul's performance in an attempt to answer the question that (briefly) gripped the world: was Paul psychic?

Anyone for tennis (and tennis and tennis...)? — Remember the match of epic proportions that took place in 2010 between the American John Isner and the Frenchman Nicolas Mahut? Just how freaky was their titanic fifth set and what odds might a bookmaker offer for a repeat?

Harder, better, faster, stronger — Using statistics to predict the Olympic medal count.

The home advantage — Using statistics to see if the host nation has an advantage in the Olympics.

Outer space: Ping-Pong is coming home — Table tennis first became an Olympic sport in 1988, but changed its scoring system in 2001 to make matches more exciting for spectators. How does the new system compare to the old one in terms of your chances of winning?

The *Plus* sports page: The curse of the duck — Why do the best cricketers in the world keep scoring zero?

The *Plus* sports page: El Niño and the Ashes — Does El Niño impact on cricket?

The *Plus* sports page: Power trip — This article looks at the tenure length of football managers and fits a model to the data.

An almighty coincidence — How do you figure out if something is really as unlikely as it seems? This article gives a hands-on introduction to statistical modelling using a real-life coincidence as an example.

Understanding uncertainty: Football crazy — On May 22nd 2009 the English Premier league had one more match day ahead, with West Bromwich Albion at the bottom of the league and Manchester United at the top, sure to remain there. Taking up a challenge from a BBC radio programme, David Spiegelhalter and Yin-Lam Ng used their statistical finesse to predict the outcome of the last matches — and they were 90% correct. Find out how they did it.

Prize specimens — In 2001 two mathematicians won £1m when it was revealed that they were the first to solve the Eternity jigsaw puzzle. It had taken them six months and a generous helping of mathematical analysis, including probability theory.

Let 'em roll — Dice are invaluable to many games, especially gambling games, but instead of playing with ordinary 1-6 numbered dice here are two interesting alternatives — with a twist!

Thomas Bayes and Mr Zootpooper — The three door problem has become a staple mathematical mindbender, but even if you know the answer, do you really understand it? Let your imagination run riot in this intergalactic application of Bayes' Theorem.

The UK National Lottery - a guide for beginners — In the early days of the UK National Lottery, it was quite common to see newspaper articles that looked back on what numbers had recently been drawn, and attempted to identify certain numbers as "due" or "hot". Few such articles appear now, and this article argues that perhaps the publicity surrounding the lottery has enhanced the nation's numeracy.

Running a lottery - for beginners — There are many different types of lottery around the world, but they all share a common aim: to make money. This article explains why lotteries are the way they are.

The coloured hat exam — This *Plus* puzzle gets you thinking about probability in terms of all possible outcomes of a situation.

Howzat! — Numbers are bandied around all the time in sports coverage — and cricket is particularly rich in statistics and rankings. To mark the occasion of the 2003 cricket world cup, this article explored what it takes to be the best.

Backgammon, doubling the stakes and Brownian motion — Backgammon is said to be one of the oldest games in the world. This article discusses one particularly interesting aspect of the game — the doubling cube. It shows how a model using Brownian motion can help a player to decide when to double or accept a double.

On the ball — If your team scores first in a football match, how likely is it to win? And when is it worth committing a professional foul? This article shows us how to use probability to answer these and other questions, and explains the implications for the rules of the game.

Outer space: a collector's piece — How many of one thing should you expect to have to buy in order to complete the set?

### Understanding life

Understanding uncertainty: Visualising probabilities — Probabilities and statistics: they are everywhere, but they are hard to understand and can be counter-intuitive. So what's the best way of communicating them to an audience that doesn't have the time, desire, or background to get stuck into the numbers? This article explores modern visualisation techniques and finds that the right picture really can be worth a thousand words.

It's a match! — At first glance it might seem that if the police have matched a suspect's DNA to evidence from the crime scene, then the case is closed. But some statistical thinking is required to understand exactly what a match is, and importantly, how juries should assess this as part of the evidence in a trial.

Why are we here? — David Sloan calculates how likely it is that our Universe exists. In this article he explains how, and why the answer can help shape our theories of physics.

Understanding uncertainty: 2845 ways of spinning risk — Would you prefer a game with a 90% chance of winning, or one with a 10% chance of losing? Of course it's the same thing, but research has shown that people's perception of risk is surprisingly vulnerable to the way it's presented. This article explores how risk can be spun and there's an interactive animation for you to have a go yourself.

Counting deaths: war as a statistical problem — How many people died? It's one of the first questions asked in a war or violent conflict, but it's one of the hardest to answer. The best we can do is come up with estimates, but the trouble is that different statistical methods for doing this can produce vastly different results . So how do we know how different methods compare?

Evaluating a medical treatment: how do you know it works? — The gold standard for testing new drugs is the randomised controlled trial. But how does it work?

Understanding uncertainty: Small but lethal — Comparing and communicating small lethal risks is a tricky business, yet this is what many of us are faced with in our daily lives. One way of measuring these risks is to use a quantity called the micromort.

Understanding uncertainty: Microlives — Many risks we take don't kill us straight away: think of all the lifestyle frailties we get warned about, such as smoking, drinking, eating badly, not exercising and so on. The microlife aims to make all these chronic risks comparable by showing how much life we lose on average when we're exposed to them.

Understanding uncertainty: Infinite monkey business — An infinite number of monkeys producing the complete works of Shakespeare is not just a probabilistic certainty, it also gives us an insight into how long we can expect to wait for a rare event to happen.

The Carol syndrome — Being attractive: is it a blessing or a curse?

Does the Iranian election stand up to statistics? — Statistical tests were used to detect potential fraud in the 2009 Iranian election, which was widely accused of being unfair.

Understanding uncertainty: What was the probability of Obama winning the election? — Some probabilistic musings with hindsight.

Understanding uncertainty: How long will you live? — An introduction to mortality statistics with an interactivity to calculate your own life expectancy.

Understanding uncertainty: Breast screening, a statistical controversy — One in nine women will get breast cancer in her lifetime, and it seems sensible to screen women for breast cancer to treat them as early as possible. But screening is a controversial issue.

The best medicine — To make hard decisions, you need hard facts. Medical statistics can help us to decide what treatment to look for when we are ill, and to estimate our chances of recovery.

The mathematics of diseases — Over the past one hundred years, mathematics has been used to understand and predict the spread of diseases, relating important public-health questions to basic infection parameters. This article describes some of the mathematical developments that have improved our understanding and predictive ability.

Keeping track of immunity —Dengue fever does the opposite of what you might expect. Unlike for many diseases, if you've had this tropical virus and recovered, you might be worse off, as a second exposure to the dengue virus can be life threatening. So keeping track of the strains of the diseases is an important problem which can be solved with the help of a little randomness.

Death and statistics — Actuarial science began as the place where two branches of mathematics meet: compound interest and observed mortality statistics. Financial planning for the future is therefore rooted firmly in the past. This article takes us through some of the mathematics involved, introducing us to some of the colourful characters who led the way.

Florence Nightingale: The compassionate statistician — Florence Nightingale survives in our imaginations as an inspired nurse. But the "lady with the lamp" was also a pioneering and passionate statistician. She understood the influential role of statistics and used them to support her convictions. So to commemorate her on the centenary of her death, we'll have a look at her life and work as a statistician.

Looking out for number one — You might think that if you collected together a list of naturally-occurring numbers, then as many of them would start with a 1 as with any other digit, but you'd be quite wrong. This article explains why Benford's Law says otherwise, and why tax inspectors are taking an interest.

Pools of blood — A biologist has developed a blood test for detecting a certain minor abnormality in infants. Obviously if you have blood samples from 100 children, you could find out which children are affected by running 100 separate tests. But mathematicians are never satisfied by the obvious answer. This article uses information theory to explain how to cut down the number of tests significantly, by pooling samples of blood.

The crystal ball — If you had a crystal ball that allowed you to see your future, what would you arrange differently about your finances? This article explores the pensions crisis, and how actuaries use statistical and modelling techniques to plan for all our futures.

Understanding uncertainty — What's the risk of passive smoking? Or climate change? How big is the terrorist threat? And should we trust league tables? These issues concern all of us, but it's not always easy to make sense of the barrage of media information. David Spiegelhalter, Winton Professor for the Public Understanding of Risk,
gives *Plus* his take on uncertainty.

Maths in the dock —Two chemists talk to *Plus* about the vital role of maths in presenting criminal evidence.

From restaurants to climate change — How linear models are used to explain diners' tipping behaviour and global warming.

Modelling catastrophes — Hardly six months go by without a natural disaster striking some part of the globe. While it's next to impossible to predict these catastrophes, let alone prevent them, mathematical modelling gives a way to prepare for their impact.

Measuring catastrophic risk — Insurance companies offer protection against rare but catastrophic events like hurricanes or earthquakes. But how do they work out the financial risks associated to these disasters?

The mystery of Zipf — The frequency distribution of words in a text follows a well-known pattern known as Zipf's law.

Tasty maths — Zipf's law again, this time in the statistical hierarchy of ingredients in cook books.

Survival odds: are all soldiers equal? — Using statistics and probability theory to understand survival rates in the army.

### Lies, damn lies

The leaning tower of PISA? — According to the 2013 PISA study Britain's teenagers have dropped out of the top 20 in reading, maths and science… Cue predictable headlines… But do the statistics add up?

Understanding uncertainty: A 113% chance of death? — A recent study from Harvard reported that eating red meat is associated with a 13% increased risk of death. But what does this mean?

It's all cricket's fault — Correlation isn't causation. If it was, then cricket would be to blame for high oil prices.

The *Plus* sports page: El Niño and the Ashes — Another cricket story, this time looking at the effect of El Niño on cricket.

Another Christmas Carol — This short piece explores Charles Dickens' negative attitude to statistics.

The tiger that isn't — NHS budgets, third world debt, predictions of global warming, inflation, Iraqi war dead, the decline of fish stocks or hedgehogs, the threat of cancer — there's hardly a subject people care about that comes without measurements, forecasts, rankings, statistics, targets, numbers of every variety. Do they illuminate or mislead? This article takes a look at numbers in the media and shows that a little maths goes a long way in unravelling dodgy media claims.

A league table lottery — How reliable are league tables? The first article in a two-part series on league tables, using the National Lottery as an example.

The Premiere League — The second article in the two-part series on league tables, using the Premiere League as an example.

Damn lies — "Lies, damned lies, and statistics..." This article tells us how to tell good statistics from bad, and make sure your cat is well-fed.

Building trust in statistics — The UK Statistics Authority holds the government to account.

All about averages — Did you know that you can't average averages? Or that Paris is rainier than London ... but it rains more in London than in Paris? This article explores the dangers that face the unwary when using a single number to summarise complex data.

Coincidence, correlation and chance — How much evidence would you need before buying into a get rich quick scheme? Do high ice cream sales cause shark attacks? And just how likely was it that you were ever born? Find out how, when it comes to probability, our instincts can lead us seriously astray.

Beyond reasonable doubt — In 1999 solicitor Sally Clark was found guilty of murdering her two baby sons. Highly flawed statistical arguments may have been crucial in securing her conviction. As her second appeal approached, *Plus* looked at the case and finds out how courts deal with statistics.

Are the polls right? — The British General Election is an example of how simple mathematical ideas help in understanding information that involves numbers.

Could maths have caught Shipman? — A news item exploring the murders committed by the GP Harold Shipman.

### Try it yourself with our sister site NRICH

Here is a selection of problems from our sister site NRICH. For all NRICH problems on stats and probability, see the NRICH archive.

Win or lose? — A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2*n* plays he has won exactly *n* times. Has he won or lost money?

Probability resources — This set of resources for teachers offers interactive environments to support probability work at Key Stage 4.

Epidemic modelling — Join in this ongoing investigation. Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

The mean game — Edward Wallace based his A Level Statistics Project on The Mean Game. Each picks 2 numbers. The winner is the player who picks a number closest to the mean of all the numbers picked.

Magical maze — Investigations and activities for you to enjoy on patterns in nature.

Football World Cup simulation — A maths-based Football World Cup simulation for teachers and students to use.

Bet you a million — Heads or Tails - the prize doubles until you win it. How much would you pay to play?

FA Cup — In four years 2001 to 2004 Arsenal have been drawn against Chelsea in the FA cup and have beaten Chelsea every time. What was the probability of this? Lots of fractions in the calculations!

Playing squash — Playing squash involves lots of mathematics. This article explores the mathematics of a squash match and how a knowledge of probability could influence the choices you make.

Snooker — A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

Snooker: frame 2 — It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

Knock out — Before a knockout tournament with 2^{n} players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

Rain or shine — Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

Over-booking — The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

PDF — Given a probability density function find the mean, median and mode of the distribution