Brief Description

This unit introduces some elementary ideas of probability through throwing dice. Contact with games such as Ludo and snakes and ladders leads some children of this age to think that throwing a six is harder than throwing any other single number. Although the expected number of throws to get a six is 6, there are occasions when it takes much longer than this. It may be that such occasions are remembered. There may also be a subconscious comparison of throwing a six with throwing all other numbers, not just one in particular. Some experiments are described to enable pupils to test this hypothesis systematically. The data obtained is rather difficult to comprehend as many figures are involved, so pictorial representation and data reduction are used to convey the main points of the distribution. Results from ordinary dice are compared with results from biased dice.

Design Time: 5 hours

Aims and Objectives

On completion of this unit pupils should be able to obtain data by direct counting, and to tabulate it. They should be able to obtain the mode and calculate the median of a small set of numbers. They should be able to draw and read a bar chart.

They will have practised collecting data, assigning probabilities by studying relative frequency, comparing two sample distributions and drawing simple inferences from distributions and reading data presented in tabular and graphical form.

They meet examples of:

1. the fact that not all samples are identical
2. the effect of size on sample distribution
3. biased and unbiased outcomes
4. independent outcomes
5. relative frequencies settling down in biased and unbiased situations.

They gain experience in collecting data under different circumstances.

Prerequisites

Previous experience of drawing bar charts would be helpful but is not essential. Pupils will need to know the meaning and order of fractions and decimals between 0 and l and use a simple scale on the vertical axis of a graph.

Equipment and Planning

Section A compares the difficulty of throwing a six with throwing a one. All the class can throw sixes and then ones, working in pairs, or the class can be asked to work in sets of two pairs. Sufficient dice for one between two pupils are required.

Section B looks at the likelihood of throwing a six with a die and investigates the connection between the 'settling down' of relative frequency and probability.

Section C covers much the same ground as Section A, but uses a die biased in favour of a six or a one.

Section D repeats Section B with the biased die.

Section A needs dice (more or less unbiased), squared paper, and pages R 1, R2 and R3.

Section B needs dice (more or less unbiased) and pages R4 and R5. Coloured pencils are helpful.

For Section C each pair of pupils requires thin card or thick paper (sheet size 9 cm x 7 cm), sellotape, or glue; or squared paper, sellotape and extra paper or card (see below).

Section D needs page R4.

Core material is Sections A, B and C, with A1(g) and C4(e) optional for more able pupils. Section D is optional, for more able pupils or those who work more quickly.

Detailed Notes

Section A

Section A begins by questioning the common belief among younger pupils that a six is harder to get than any other number. The experiments have two purposes. One is to see how long it takes to throw a particular face on a die.

The other is to compare the results of throwing for one face with the results of throwing for another. Pupils complete bar charts and draw some inferences. It is useful to point out that not all samples are identical and to indicate the effect of size on sample distribution. The choice of 25 times for the experiment (i) eases the calculation of the median, and (ii) is large enough to give a reasonable sampling distribution.

The theoretical distribution for large numbers of throws of unbiased dice has:

mode = 1 median = 4 (The mean, not calculated here, is 6.)

It may be possible to get some of the brighter pupils to see that their experiment can be considered as a sample from an infinite population of throws to get a six, but this is not mentioned in the Pupil Unit.

It is fair for all pupils to have to throw a six before starting at Ludo since it applies to all players, and they use the same die. To have all players starting together would change the nature of the game, and possibly take away some of the fun. It is a matter of opinion which way is better. Some examples of other ways of starting are: tossing coins, pulling names from a hat, drawing straws and drawing cards.

A1
Unbiased dice and page Rl are required. It is not easier for Ann to start by throwing a one, provided that the die is unbiased.

It is suggested that the class work in pairs and then double up with another pair. Half the class are throwing sixes and the other half are throwing ones. They then compare results. If time is not a problem, each pair could throw both ones and sixes.

When copying the other pair's results in (c), it is the total number of throws, not the actual results, which is important. Usually the smallest number of throws taken to get a six or a one is 1.

The last few optional questions raise the ideas of independence and that there is no theoretical upper limit to the number of throws that could be taken. More able pupils can be asked to look at their sequences to see if their answers are carried out in practice.

A2
Page R2 is needed. R2 allows for 30 throws to get a six or one. If more throws are taken, the chart will have to be drawn out separately or, a less acceptable alternative, by inserting an extra bar at the right-hand end, labelled 'More than 30 throws.'

A4
The distributions should lead to the conclusion that the difficulty of throwing a six and a one are about the same. The two medians should be approximately equal. There may be disagreements if medians are very different. There is a possibility of discussing the effect of sample size and the need for more results. This simple question is of the essence of statistics. We are trying to use samples to draw conclusions about populations (or distributions).

A5
Squared paper is needed. If the effect of sample size is not discussed in A4c, it could be brought in here. The results from throwing sixes and ones (25 results per pair) need to be collected. How this is done is a matter of individual preference. The data could be collected, for example, as written in Table 1 (i.e. at random) or as written for the calculation of the median. In either case the use of a blank table on a blackboard can speed collection. The method chosen depends on the capability of the class at handling larger quantities of data. It will probably be worth a final count that the correct number of results has been collected.

1. Half the class could draw the chart for throwing a six, the other half the chart for throwing a one. The class modes and medians should now be nearer the theoretical values quoted above. A six and a one should be equally hard to get.

Section B

Section B involves throwing a die again. Section B2 could use the first 20 results of Table 1. The connection between relative frequency and probability is introduced. The 'settling down' of relative frequency in these circumstances is fundamental to the idea of using relative frequency as a basis for estimating probabilities. The graphical method chosen is based on an idea used in the D.I.M.E. project of G. Giles of the University of Stirling, whose material is published by Oliver and Boyd.

The advantages of this form of representation are as follows:

1. pupils do not need to keep a count of the number of throws,
2. no tallying is needed,
3. information is stored in sequence for later use if needed,
4. pupils do not have to calculate fractions or change fractions to decimals,
5. it can be used for a wide variety of experiments that have success and failure as the two outcomes,
6. the idea of a limit is present. If pupils see graphs on transparencies, then, though they are wildly different at the beginning when the transparencies are overlaid on an overhead projector, they show a trend towards the same results.

A six is expected once every six throws. If unbiased, each face of the die is equally likely to land on top. The probability of any particular face being on top is ¹/₆ since it is one of six equally likely cases. Children's intuition (or just guessing) may give different answers here. Too much initial discussion is probably best avoided, since those who 'know' the answer may spoil the discovery by those who don't. The rest of the section should provide answers.

B1
Page R5 and unbiased dice are required. Coloured pens or pencils help to show the paths more clearly.

The idea is to start where it says Start and move to the right for a success or to the left for a failure and see where your route finishes. Down the left-hand side the number records how many throws; the number of points in from the left gives the number of successes. If you reach an edge of the graph, the next part of the path may be vertically downwards rather than to the left or right. When turned through 90^o, the graph becomes the usual plot of relative frequency (y-axis) against number of throws (x-axis). This can be seen as the first part of the graph.

It is hoped that 40 throws will be seen to settle down better. A larger sample size usually shows the settling down of relative frequency more clearly.

The path should settle down at about the 0:5 level for fraction of successes.

There is an equal likelihood of throwing an odd or an even number with an unbiased die, so we have a theoretical probability of ¹/₂.

B2
Page R4 and unbiased dice are needed. Coloured pencils are helpful. In a the fraction should be given as 'so many out of 100'. You can expect about 17 sixes, and the paths should settle down at about the 0.17 (i.e. 1 in 6) level.

There are six faces on a die. With an unbiased die each face is equally likely to land on top, so we can expect a six to turn up 1 time in 6. The theoretical probability of throwing a six with an unbiased die = ¹/₆.

It is recommended that some of the class try to throw a one.

Section C

Section C covers the same ground as Section A, but uses biased dice.

A discussion on the concepts of biased and unbiased might be of value. The statistical ideas are the same as those of Section A, but with the added emphasis on dice biased in favour of one or six.

C1
The recommended method of making a biased die is given in the Pupil Notes. An alternative method is given here:

Draw the diagram on squared paper, using squares with sides 2 cm.

Mark the numbers 1 to 6 on the BACK of the squares shown. Use sellotape to stick extra paper or card on the front of the square marked *. Cut out along all the solid lines. Fold along the dotted lines. Fold the square marked F in front of the square marked with a 6 on the back. Let all three strands point towards you. Plait each strand under and over each other until finally the tab tucks into the cube. The cube should now look like a die and be biased in favour of a six.

This method does give a cube which stands up to the rigours of throwing.

Neither die is easy to roll. They should be spun in the air like tossing a coin. Dice already biased in favour of different faces are available from E.S.A.

It is suggested that each pair be allowed to choose the bias of their own die, and then exchange their die with another pair who have to guess the bias. Very able children might use dice biased in favour of other single numbers such as two or three.

The net for the die could be reproduced using a spirit master.

C2
A six is expected to show up more quickly, therefore fewer throws are needed to get a six. There will be more squares towards the left-hand side of the chart.

C3

2. No R page is provided for this chart. Squared paper is required. The bias of the die could mean either very many throws or very few throws needed to get a six.

C4
The shape of the new chart should differ from that of Figure l. If the die is biased in favour of six, then fewer throws tend to be needed to get a six. If the die is biased in favour of one, then more throws tend to be needed to get a six. The mode in each case should be about the same: in all cases its theoretical value is 1. The median should change with the bias of the die. If the die is biased in favour of six, then the median is probably lower. If the die is biased in favour of one, then the median is probably higher.

The pupils' answers to c should determine the bias.

5. Yes, you could play Ludo with a biased die. It would be easier to get started and there would be faster progress on the board. It may be harder to get home at the end (fewer ones and twos).

*Section D

Section D repeats Section B but uses biased dice.

Page R6 and biased dice are required. Coloured pencils are helpful. This section should show that biased dice lead to relative frequencies that settle down in just the same way as unbiased dice. The techniques involved are the same as Section B.

With a die biased in favour of a six, the probability of throwing a six is higher than ¹/₆.

With a die biased in favour of a one, the probability of throwing a six is less than ¹/₆.

A die biased in favour of a six will land with 6 on top more often than an unbiased die. A die biased in favour of a one will land with 6 on top less often than an unbiased die.

Answers

A1	a	No
	f	1
	g	No, not if the die is shaken properly
	h	It is not possible to say.
A4	a	Bigger
	b	About the same
	d	No
B1	b	9, 9/20 or 0.45
	c	4,4/10 or 0.4 (Beware of the temptation to read '8' on the horizontal axis for the first answer.)
C2	a	Easier
	b	Usually fewer
	c	Higher frequencies at the left-hand end
	d	Lower

Test Questions

1. The 29 boys in Year 2 of Park School gave their shoe sizes as:
   3, 1, 4, 5, 7, 2, 4, 5, 3, 6, 1, 8, 5, 3, 2, 4, 1, 7, 5, 6, 2, 4, 3, 4, 4, 7, 5, 6, 4.
   1. Draw a bar chart of these figures.
   2. Write down the mode of the boys' shoe sizes.
   3. Put the 29 shoe sizes in order and find the median of the boys' shoe sizes.
2. 
   
   Figure 1 - Girls shoe sizes. Year 2 Park School.
   
   Figure 1 shows the shoe sizes of the 29 girls in Year 2 of Park School.
   1. How many girls had shoe size 5?
   2. Suppose that in general boys have smaller feet than girls. Will the median of boys' shoe sizes be: larger than, about the same as, or smaller than, the median of girls' shoe sizes?
   3. Find the median of the girls' shoe sizes in Year 2 at Park School.
   4. Do the boys in Year 2 at Park School have larger or smaller feet than the girls?
3. Ian is throwing a drawing pin. If it lands like this , he calls it point and counts it a success. If it lands like this he calls it flat and counts it a failure. Figure 2 shows the results of 20 throws.
   1. How many points were there in the first 10 throws?
   2. How many points were there in the 20 throws?
   3. Copy and complete the following sentence: The probability of throwing a point on Ian's drawing pin is about: _____
   4. From these results which of these three statements seems most likely to be true?
      Ian's drawing pin is biased in favour of flat.
      Ian's drawing pin is biased in favour of point.
      Ian's drawing pin is unbiased.

   In the next 20 throws Ian obtained 15 points and five flats.

   1. e Use these results to get a better estimate of the probability of throwing a point.
      Why is it a better estimate than the one in 3c?

   

Answers

1	b	4
	c	1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,7,8. Median is 4.
2	a	2
	b	Smaller than
	c	Median is 3
	d	Larger
3	a	6
	b	13
	c	13/20, or 0.65
	d	Biased in favour of point
	e	28/40, or 0.7 i.e. combines the two sets of 20 results together. Better because the graph settles down with more throws.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 1)

Being Fair to Ernie
Wheels and Meals
Practice makes Perfect
Leisure for Pleasure
Probability Games
Tidy Tables

Units at Other Levels In the Same or Allied Areas of the Curriculum

Level 2

Fair Play
Seeing is Believing
On the Ball

Level 3

Net Catch
Cutting it Fine

Level 4

Choice or Chance
Testing Testing

This unit is particularly relevant to: Mathematics, Social Science.

Interconnections between Concepts and Techniques Used In these Units

These are detailed in the following table. The code numbers in the left-hand column refer to the items spelled out in more detail in Chapter 5 of Teaching Statistics 11-16.

An item mentioned under Statistical Prerequisites needs to be covered before this unit is taught. Units which introduce this idea or technique are listed alongside.

An item mentioned under Idea or Technique Used is not specifically introduced or necessarily pointed out as such in the unit. There may be one or more specific examples of a more general concept. No previous experience is necessary with these items before teachingWe unit, but more practice can be obtained before or afterwards by using the other units listed in the two columns alongside.

An item mentioned under Idea or Technique Introduced occurs specifically in the unit and, if a technique, there will be specific detailed instruction for carrying it out. Further practice and reinforcement can be carried out by using the other units listed alongside.

Page R1

	Numbers showing on the die	How many throws to get a six
1st time
2nd time
3rd time
4th time
5th time
6th time
7th time
8th time
9th time
10th time
11th time
12th time
13th time
14th time
15th time
16th time
17th time
18th time
19th time
20th time
21st time
22nd time
23rd time
24th time
25th time

Table 2 - Throwing a six (ordinary die)

Table 3 - Throwing a one (ordinary die)

Page R2

Figure 4

Table 4 - Throwing a six (biased die)

Page R3

Figure 5 - Throwing a six - Class results.

Figure 6 - Throwing a one - Class results.

Page R4

*Figure 9 - Throwing a six with an unbiased die
*Figure 10 - Throwing a six with a biased die

(*Cross out the one that does not apply)

Page R5

Figure 7 - Twenty throws of a die - odd or even.

Figure 8 - Forty throws of a die - odd or even.

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