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

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.

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.

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

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.

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.

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.

Lies, damn lies

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.

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 2n 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 2ⁿ 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