Probability Experiments in the

Secondary School

L. W. GATES

Probability theory is taught as part of the normal curriculum in most secondary schools, but it is not usually possible, due to shortage of time, lack of equipment, or teacher reluctance to perform probability experiments. I have been fortunate to be able to spend one period per week doing probability experiments with the first year (11 year olds) at Gillingham Technical High School.

The opportunity has arisen really as a bonus in our efforts to raise the level of attainment in basic arithmetic of the first form intake. Gillingham Technical High School is a coeducational school with an annual intake into the first form of 150 children selected according to ability. Even a selective school intake contains some children whose grasp of basic skills in arithmetic is decidedly shaky; these need their confidence built. Since practice exercises would simply bore the majority of the children we have used the opportunity to provide topics for the brighter pupils which some of them could develop privately.

An extra period was given to the first forms in which those children who needed it could be given extra tuition in arithmetic. The remainder of the children were divided into five equal groups to do work which did not form part of the normal mathematics syllabus. These five groups go to five different teachers in turn, each for six periods, one period per week for six weeks. The five topics which we are currently offering are: Probability experiments, Model making, Computing, Topology and Estimation.

I have approached my topic of Probability experiments by using work cards. There are in all sixteen experiments. The instructions for each experiment are set out on cards which I have devised myself. Each child works on his or her own at an experiment, recording the results on paper. I have found that most children complete from four to eight experiments in the six periods­some experiments are much longer than others. Some children find the recording of results in a neat, orderly fashion difficult; but, with guidance, most of them have produced some reasonably presentable work, and working independently in this way is certainly good practice for the future.

Briefly, the sixteen experiments consist of: six experiments involving the throwing of coins or dice or both, one experiment using a special die with two red, two blue and two yellow faces (The Wasps experiment), two experiments involving the drawing of beads from a bag, one experiment involving the throwing of drawing pins and another the throwing of matchsticks, one experiment using playing cards, a statistical experiment to determine the most commonly used letter of the alphabet, an experiment using a roulette wheel (the Raindrops experiment), an experiment involving the moving of counters on a squared board according to the throw of a coin, and finally a cricket game involving the rolling of a hexagonal prism­an adaptation of the game ‘Owzhat’. Most of the experiments are duplicated and the last two mentioned are for two children at a time.

I have made some of the experiments up myself­some, for example, coin tossing, are standard. I have got ideas for others from the book Luck and Judgement published by Chatto and Windus for the Schools Council. The article Probability in the Junior School by Philip Sherwood in the May 1978 edition of Mathematics in School was the source of the two highly popular experiments: the Wasps and the Raindrops.

As examples, here are the complete instructions for two experiments:

Experiment eleven

Draw a picture of a ladder with 20 rungs. Starting at ground level you wish to climb to the top of the ladder. Use a matchstick or something similar to mark your position on the ladder.

To move up or down the ladder you throw a die. If you throw an even number you move up the ladder that number of rungs; if you throw an odd number you move down the ladder that number of rungs (if you get to the bottom of the ladder and cannot go any further then you stay at ground level).

See how many times you need to throw the die in order to reach the top of the ladder. Do the experiment ten times and find the average number of throws.

How many throws do you think it would take if the ladder had only ten rungs? Try it.

Experiment five

If you toss two dice many times and add together the two scores each time, which total score do you think you will get most often? What are the highest and lowest total scores possible? Which score do you think will turn up least often?

Do the experiment and record all the scores­do the experiment 100 times. Summarise your results in the form of a table:

 Score Tally Frequency Total 100

Can you think of an explanation for your results?

Here is a selection of the responses given by the children to the questions in experiment five:

Maria The lowest score is 2. The highest score is 12. Least often score 1, most often score 7.

There is only one way you can get 12 and there is a lot of ways you can get 7.

David I think the most common score will be 6

I think the least common score will be 1.

Chi I think~I will get 7 most often. 2 and 12.

There are more ways to make 7 than any other number by two dice.

Tansie I think I will get the answer 4 most, the highest score is 12 and the lowest score is 2. I think the score I will get least often is 12. With two dice you very rarely get 12 or 2

Michael Most 7, least 12

No I cannot think of an explanation.

Stephen I think when I throw the two dice I will get the number 7 most often. The lowest score possible is 2 and the highest is 12. The total I think will turn up least often is 2.

The numbers in the middle of 2 and 12 are much easier to get than the numbers nearer the outside.

James I think 9 will come up the most and 12 the least. 2 is the lowest score, 12 is the highest score.

The more combinations available makes that number more easy to throw with dice.

It is interesting to note that some children think the least often score is 1 even though, in some cases at least, they have already noticed that it is an impossible score.

This is the second year that we have run these courses at our school. After the first year I did a survey with the last group I took to try to discover how they had enjoyed what they had done with me and to get comments about the experiments. The general reaction was very favourable and some of the comments made were valuable. As a result I have rewritten or replaced some of the experiments and amended others. There are still improvements that could be made, but on the whole, the exercise has been successful in filling a very important gap; it has also been enjoyed by the children and by myself.

Gillingham Technical High School