# For teachers

Probability --- coins experiment --- coins theory --- dice experiment --- dice theory --- for teachers

Probability is studied in mathematics using formulae. But it may help when introducing probability to do some experiments, to see if the real world matches the formulae. The easiest experiments involve tossing coins to see if they land heads up or tails. Each throw is independent, and only has two possible states, which makes the mathematics easier. However, it is extremely tedious to toss enough coins to make one worth-while experiment, let alone several to compare, quite apart from the riot caused by a class-room of children all trying to toss coins at once, dropping them, letting them roll under furniture.... So this webpage throws the coins for you.

There is an explanation of the theory of coins probability here. When you try the experiment, you will find that these percentages don't always happen. You can get figures close or even exact, but other times, you get figures quite wildly out. In fact, for 4 coins sometimes one or three heads column is higher than 2 heads (which should be highest). It is important that the children realise that probability does not tell you what the coins are going to do. It shows roughly what you can expect, but not precisely. You will also find that a small number of throws will produce more unpredictable results. Unfortunately larger numbers of throws can take more time, and if you have a slow connection it might be hard to use them. Experiment to find out which is the most suitable for you.

If you are a teacher, how you use the page is up to you. You could either use it before introducing the mathematics, to introduce both the variability, and the tendency for the coins to behave in a certain way. You could encourage them to work out for themselves that there doesn't seem to be the same probabilities for different numbers of heads (although they won't put it that way, of course). They might notice that the bar chart seems to be fairly symmetrical, with the probability for no heads being roughly the same as the maximum number of heads. Encourage them to toss the coins quite a few times, so they get a 'feel' for what tends to happen. Then they might try jotting a few numbers down, to see if they can work out what a 'typical' figure is for a certain number of coins and a certain number of heads. Perhaps different groups could concentrate on different coins/heads combinations. You could ask them if the chance of no heads is the same for one coin, two coins, three coins and four, and if not, does the chance go up or down? Once you have explored the experimental data thoroughly, you could then move onto a more formal, mathematical approach, and work out what the figures should be.

Another approach would be to handle the formal work first, and then try the webpage to see if it agrees with the theory. Here the variability might surprise them, since the simple theory doesn't mention the variation of results in the real world! You could get them to toss the coins several times, noting down the figures each time, and then analyse the figures, seeing how close the theory matches. It might be possible to bring in some IT work here, since this is an obvious candidate for a spreadsheet. Extra bright children could be encouraged to subtract the actual figures found from the predicted figure, and think about that. How wide is the possible variation? How often does it appear? You could plot a chart of this variation. They could also look at whether increasing the number of throws improves the variation, and by how much.

The last approach is to use this webpage as a bribe! "Do the serious work on probability, and you get to play on the computer tossing coins."

By the way, the percentages don't necessarily add up to 100%. This is because of rounding errors. Please tell any child that notices this that they get a pat on the back from the website writer.

If you want to move onto more complicated probability, then there is a dice experiment, and the theory behind dice probability.