What's the chance of another red car, Miss?
People are always trying to predict the future, with varying degrees of success. They may consult the stars or use elaborate-looking charts, but more often than not they rely on intuition, premonition, or a guess.
We discover from an early age that the world is ruled more by chance than by certainty, and this is where probability comes in - working out the chance of an event taking place. Today, many people make a living by predicting the outcome of an event but, with the possible exception of Mystic Meg, from weather forecasters to racing tipsters and insurance company staff, they base their judgments on carefully recorded, observable and verifiable data.
Young children will already be familiar with the terms "fair" and "unfair" in the games they play, and this is a good starting point for introducing the topic of probability. For instance, why does a football referee toss a coin at the start of a match? Is this a fair way to start the game?
Many practical activities can help teach probability. Coins are perfect for introducing the concept of "fair", "evens" or "equally likely". For instance, ask children to toss a coin 50 times and record the result, first as a tally chart and then as a graph.
It may also be a good idea to make a graph of the whole class's results. Ask the children if they can predict the outcome of the next toss of a coin if the previous nine have all been tails. Eventually the children can be introduced to the concepts of a one in two, or 1/2 chance of getting a head or a tail. The same exercise can be repeated with anything that has equal odds, such as a cube with three sides in one colour and three in another.
To develop the idea, use a tetrahedron to introduce odds of one in four, or 1/4. Or have two sides blue and two sides red, giving odds of 1 out of 2. Making spinners, divided into any number of equal parts, is also a good design and technology exercise.
Another way of developing the idea is to use everyday terms. Draw a line and mark positions from no chance (0 probability) on the left, through poor chance, even chance (0.5 probability) and good chance to certain (probability of 1). Children can then indicate where on the line certain statements, such as "I will eat today", "I will die one day", "If I toss a coin it will land heads", "England will win the next World Cup" and "It will snow in London on June 24 next year", will fall. Some of the more contentious statements will promote lively discussion.
You can adapt the exercise, using a die to work out the likelihood, for instance, of throwing a six, an odd number, a seven, a number greater than one or a number between one and six.
Another exercise using coins involves asking the children to toss two coins at the same time. They can guess the number of times they would throw two heads, two tails, and one of each in 60 goes and then check practically. Many will predict 20 from each category, and will be surprised by the results.
Next, they can use four coins. To stretch the bright pupils, ask them to list all possible combinations. Explaining what it means to work "systematically", the teacher can get the children to write HHHH, HHHT, HHTT and so on. By the end of the exercise, they will have 16 possible combinations and can work out that the chance of four heads is the same as the chance of four tails - one in 16, the chance of two heads and two tails is six in 16, and three heads and a tail is the same as three tails and a head at four in 16, or 1/4.
Dice are excellent for probability work. Ask the children to roll one dice 50 or 100 times. Ask what are the chances of rolling, say, a two or a six?
The real fun starts with two dice. One of my favourite games is called pick-a-track, which can be played with between two and six pupils. Draw a simple squared grid 12 squares wide by however many you like (depending how long you want each race to last). Then number each square along the bottom from one to 12, (the one can be omitted if you like). To move a horse roll two dice and add the two numbers. If the total is, say, five, move a counter (the horse) one space in the track above the number five. The winner is the first to reach the end of the grid.
Eventually, the pupils will notice certain tracks win more often than others. But why? Again, ask the bright pupils to work out all possible combinations of the two dice, and the reason for the dominance of horse number seven soon becomes apparent. The number seven has more possible combinations than any other number - six out of 36, or 1/6.
Working with two dice provides a good opportunity to talk about sampling. Ask the children to roll two dice 100 times, and record their results as a graph. Which number comes out the highest? Obviously, chances are it will be seven, or maybe six or eight. But this would not necessarily follow if the dice were rolled only 10 or 20 times. In other words, you need a fair sample, and the more times you roll the two dice, the more likely the graph will look like the expected distribution (figure 1).
A further extension is to roll the two dice 100 times again, but this time record the difference between the two numbers. For instance, five and three give two, four and four give zero. In fact, zero is more likely than any other result.
Two further practical applications using probability are vehicle surveys and designing games for school fetes. Ask the children to record the number of red cars that pass by (or near to) the school out of 20 consecutive vehicles. Ask them to predict the number of red cars in the next 100 or 200 vehicles, and then see how near their predictions are. Ask them which is the most popular car colour, and do the car-makers need to have this information when deciding which colour to paint their cars?
Pick-a-straw is a popular game at fetes. People are invited to take their pick from a tin full of straws. If they pick a coloured one they win a prize. Devising such a game with various numbers of plain and coloured straws provides a good assessment of pupils' knowledge of probability, and gives them the chance to apply their knowledge.
Tell the pupils the main aim of the game is to make as much money for the school as possible. They will soon see there are many factors to consider and discuss. These include the cost of playing, the chances of winning, the amount of prize money, the profit margins and, finally, the market research and trialling.
The need is to find a balance between maintaining people's interest by having a fair number of winning straws, keeping the cost of playing reasonable and having a suitable amount of prize money and ensuring a reasonable profit margin.
Pupils can make a chart showing variations of cost of playing, number of winning straws, chances of winning a prize, cost of prize, income after 100 goes without a payout, expected payout after 100 goes, and profit (income minus expected payouts).
By following these activities you should cover the three sections on understanding and using probability in the national curriculum - and ensure you have much fun and discussion on the way.
Jon Swain is a former mathematics co-ordinator and deputy head of a primary school in Essex