A new way of recording relative frequencies for simple experiments.
Use of the Recording Sheet
Automatic Counting
Percentage of Wins
Relative Frequency
Visible Trends
Will They Cross?
Estimating in Advance
Construction of the Recording Sheet
Pascal's Triangle
Binomial Coefficients
Probability when p = 1/2
Confidence Limits
Distribution of Means
Ten Wins in a Row
Independent Events

Classroom experiments in probability tend to be avoided by most teachers of mathematics. They are seen as involving problems of organisation and control which outweigh any advantages they may bring.  Normally the teaching objective is the ability to answer examination questions on probability and, time being short, practical work is seen as a non-essential luxury.

My experience leads me to believe that pupils and students are unlikely to develop an understanding of probability without an appropriate background of experimental experience.  It was this that led me to develop the set of 30 experiments using 'shakers' that has been published for DIME Projects by Oliver and Boyd. My objective was to design material that would enable pupils working in pairs to carry out 100 trials each and write up the experiment within the space of a period.  Pupils at the bottom end of the secondary school were mainly in my mind, and that is why the word probability occurs only in the title.  Those not yet 'ready' for probability can treat the experiments as guessing games.  Hopefully the fact that others in the class believe they can anticipate results that are just a matter of chance' will lead to discussion and the development of the basic concept of probability. I have found the material to be of considerable value also with students and teachers who have often gained a new insight into probability through working with it.

One important part of this development work was the creation of a recording sheet that would be both easy to use and effective in promoting insight and understanding.  The result is shown in the diagrams, and it is the purpose of this article to describe some of the advantages that can come from using it.
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1. Use of the Recording Sheet
To record the sequence of 'wins' and 'losses' a continuous line is drawn from the starting point, preferably using a felt-tip pen.  This line always follows one of the thin lines printed on the sheet, and always moves down the recording sheet.  At each circle there are two possible lines that can be taken, and the choice depends on the result of the trial: WIN-go to the right, LOSE-go to the left.  Diagram 1 shows the sequence starting W W L L W. The actual size of the sheet is A5.

Once the pupils understand how to use the sheet, the recording of their results becomes simple and quick.  Normally pupils work in pairs, one shaking while the other records.  Although the speed of shaking and calling out the result sometimes reaches 25-30 trials per minute, the recorder can easily keep up with this pace.
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2. Automatic Counting
The counting of the trials is done automatically by the recording sheet.  When the pupil reaches the bottom line he has recorded 50 trials.  What is more, he can at once read off the number of wins in this set of 50, from 0 at the bottom left corner to 50 at bottom right.  The 50 trials recorded in diagram 1 include 28 wins.  A second trace on the same sheet (preferably using another colour) makes up the total of 100 trials.
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3. Percentage of Wins
In addition the pupil can judge the percentage of wins not only at the end of the 50 trials but also at any earlier point using the vertical lines.  For example, the point X on diagram 1 lies on the line indicating 60%.
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4. Relative Frequency
Another aspect is that no division is needed in obtaining relative frequencies.  The geometry of the diagram allows one to bypass the arithmetical computations that so often act as a barrier to the understanding of probability we wish the pupils to develop.
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5. Visible Trends
Another important result of this method of recording is that the trend in the results is visible as the experiment proceeds.  Further, as the trace moves down the recording sheet during each set of 50 trials the movement from side to side diminishes quite obviously.
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6. Will They Cross?
When a second trace is drawn on the sheet to record the second set of 50 trials a surprising degree of emotional involvement can develop in adults as well as in children.  Will the second line cross the first?  This I had not anticipated, but I have no doubt that personal interest of this kind is helpful in developing an intuitive feel for the basic concepts involved in probability.
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7. Estimating in Advance
Quite deliberately the Report page on which the pupil writes up the experiment requires her to write down, before the experiment is started, how many wins she expects to get.  This leads to an increased emotional involvement in the experiment, because she then wants the result to be close to her estimate.  When running workshops for teachers I illustrate this point by using one particular experiment.  This involves two pairs of balls in a cylindrical tube.  A TRIAL involves shaking the tube and placing it upright on the table, so that the balls come to rest either in a crossed pattern (diagram 2a) or side by side (2b).

It is very tempting to assume that these outcomes are equally likely, and indeed most pupils, students and mathematics teachers do estimate that they will WIN (diagram 2b) half the time.  Unfortunately for them the theoretical probability is 2/3 as can be seen by considering diagram 2c.

 If one black ball comes to rest in position A, then of the other three spaces that the second black ball could occupy two give a WIN.  When carrying out the experiment however most people expect, and want, the line to come down near the 50 % mark in the middle of the page.  When the trace shows a tendency to wander over to the right this can cause considerable worries, especially when the second trace confirms the inclination of the first. I have been accused of providing biased shakers, and I heard one teacher at an Australian workshop exclaim to his partner that he had 'lost all faith in the laws of probability'.  Of course there were other teachers who accepted the discrepancy apparently without worry, which raises the important question: When should one start rethinking one's estimate?
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8. Construction of the Recording Sheet
Let us now look at links between the construction of the recording sheet and other topics in mathematics at a variety of levels.  Firstly it should be noticed that each row of small circles divides the width of the recording sheet into equal parts.  This is why, for example, point X on diagram 1 is 3/5 of the way across the page and thus on the 60%. vertical line.  It follows that if the sheet is placed on its side (which in fact has certain advantages for recording purposes) the curves are of the form y = k/x and y = h - (k/x).
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9. Pascal's Triangle
Returning to point X on diagram 1, we can see that this point represents the situation where we have had five trials which include three wins.  We can ask in how many ways we could reach this position.  This can be looked at in terms of different sets of results: W L W L W,  L L W W W,  L W W W L, etc.  Alternatively we can consider the number of different paths from the starting point to X. Either way we get the answer 10. If we work out the corresponding numbers for the other points on the same level as X we obtain the binomial coefficients 1 5 10 10 5 1 which one finds in the expansion of (x + y)⁵.
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10.  Binomial Coefficients
A natural continuation of this line of thought leads to questions such as: How many paths are there from START to the centre circle on the bottom row?  Calculation by extending Pascal's Triangle row by row downwards would be exceedingly tedious, but here is a ready made opportunity to invoke, or develop, the formula for the binomial coefficients which gives the expression 50.49.48.47.46 .... 27.26/1.2.3.4.5 ... 24.25

At this point any careful and enterprising pupil with an electronic calculator and the necessary time and interest can work out that the answer is approximately 126 000 000 000 000.  The linking of different aspects of mathematics in this way to a familiar recording sheet that pupils have used in experimental work, and to basic questions like: How many paths to X? must surely make such mathematics more accessible and meaningful to students of all ages.
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11. Probability when p = 1/2
Taking the special case where WIN and LOSE are equally likely, it follows that all paths, down to the X level say, are equally likely.  We note that the total number of possibilities (paths) for the first five trials is the sum of the numbers on the fifth row of Pascal's Triangle after the START, which is 2⁵ or 32.  Thus the probability of a path passing through X is 10 out of 32, or 0. 3125.  If we now consider paths right down to the bottom line of the recording sheet we see that the total number is 2⁵⁰ = 1 126 000 000 000 000 approximately, and it follows almost at once that in an experiment in which p = 0.5 one should expect about half the zig-zags on the recording sheets to reach the bottom line in the range 23 - 27 wins, inclusive.  The probability of hitting the dead centre is about 0.1.
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12.  Confidence Limits
In essence the above discussion provides a simple introduction to the sophisticated concept of confidence limits.  For classroom purposes it might be appropriate to take confidence limits of 90 %. These can be shown on the recording sheet (diagram 3) not just a range on the bottom line but as two curves between which, at any level, you would expect 90 % of the zig-zags to lie.  It is clear from the appearance of these curves that the more trials you make the closer you can expect to come to the theoretical value of the probability.

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13.  Distribution of Means
While the Probability Kit is designed to allow pairs of pupils to work on different experiments at the same time, there are advantages in carrying out class experiments.  One major one is that the data can be collected on the blackboard for discussion purposes.  This provides a distribution of results that can be considered at various levels.  Firstly it can be related to confidence limits as discussed above.

Secondly, the mean for the class can be calculated. In one workshop in which I used the experiment described in section 7, the information was volunteered that the mean number of wins per 50 trials was 33.33. This enabled me to ask what 'accuracy' we should expect from the mean, bearing in mind the number of people taking part.  If a workshop involves 40 people each recording two lots of 50 trials, within what range should one expect the class mean to lie?
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14.  Ten Wins in a Row
A simple and important point about this method of recording that has not been discussed yet is that it retains the order in which the results came.  One advantage of this is that it is possible to ask the class if anyone has recorded ten wins in a row.  This is very easy to check, whereas normal methods of recording do not retain such information.  One can also ask: How many cases of ten wins in a row would you expect there to be in a class of this size?  Here is another opportunity to discuss more advanced problems in probability.
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15.  Independent Events
Another advantage cropped up unexpectedly during the particular workshop experiment described earlier.  One frustrated student complained that the reason he was getting too many wins was that the tube was too narrow and the balls couldn't pass each other easily.  This meant, he said, that a win was more likely to be followed by another win and this was why he was getting 'too many wins'.  The obvious response was to ask if he found that a loss was more likely to be followed by another loss.  A glance at his sheet showed him that this was not the case and he realised his hypothesis was untenable.
 

This article is based on material in the final report of a research project funded by the Scottish Education Department: School Mathematics Under Examination.. 3 Some factors affecting the learning of mathematics.

Reference
1. Probability Kit No. 1. Oliver & Boyd ISBN 0 05 003050
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