Brief Description

Starting with the idea of fairness, this unit develops a method of random sampling using random numbers, which is then applied to the selection process behind premium bonds.

The premium bond situation is modelled by a simple simulation game Ernie in which some associated flaws and fallacies are studied.

Design Time: approximately 4 hours.

Aims and Objectives

On completion of this unit pupils should be able to draw up their own random number tables using a suitable randomizing device, and to use random number tables to draw a small sample from any population up to 100 whose members can be numbered.

They should be more aware of the connection between random selection and equally likely outcomes, some of the problems in making numbers truly random and some phenomena which exhibit random variation.

They should understand the statistical basis to premium bonds, and they will have practised reading and interpreting data presented in tabuhr and bar chart form.

Prerequisites

For one part of B1, which can be omitted, pupils need to be able to use a pair of compasses and a protractor and to understand the words radius, radii and polygon. Optional questions require the drawing of bar charts, and it will help if pupils have previously drawn simple bar charts.

Equipment and Planning

Approximately 200 plastic counters or beads are needed in Section A and 100 in Section C. Coins and dice are needed in Section A, together with squared paper and a container (a bag) from which to draw numbers. Compasses, ruler, protractors, card and a 'nail' and scissors are needed in Section B as well as dice and coins. One 10 by 10 square would be useful in Section C for recording the purchase of premium bonds, numbered 0-99.

Individual working is possible in Sections A and B Section A uses easy games with dice and coins to establish the ideas of fairness and equal likelihood. The last two games introduce the basic idea of random numbers and random sampling. Section B develops the ideas of random numbers more fully and leads to the more formal use of random number tables. Section C uses random number tables to show how Ernie selects prize-winners from premium bond holders.

Detailed Notes

Section A

Pupils need to be grouped in threes for all games in this section. (It may be useful to give names Ann, Brian and Charles to the members of each group for all the games.) It is recommended that each group plays either A1 and A2 or A3 and A4 and then moves on to A5, A6 and A7.

It should be emphasized that if each has an equal chance of winning it is a fair game. It is not a question of whether or not it is possible to cheat.

The fairness or unfairness should show up in the results, though allowance must be made for random variation. Collecting chss results should dispel any doubts about fairness. The bar charts are optional though they can provide visual evidence of fairness or bias.

Theoretical explanations which involve looking at equally likely outcomes may be given to brighter pupils.

A1
A FAIR game.

Theoretical results come from:

A2
An UNFAIR game.

Suppose Brian always calls Heads against Ann, and Charles always calls Tails. Then theoretical results come from:

Ann and Brian have a 50% chance of being eliminated at the first throw (semi-final). Charles cannot be eliminated until the second throw (final). Charles has probability ¹/₂ of winning; Ann and Brian have probability ¹/₄ each of winning.

A3
Assuming an unbiased die, this game is clearly FAIR.

A4
An UNFAIR game.

Brian, choosing Head and a Tail, has more chance of winning. Theoretical results came from:

The three outcomes double Head, Head and Tail, double Tail are not equally likely. The results of the game should emphasize this.

A5 and A6
Between them these two games involve the ideas behind random numbers. A5 involves numbers but not randomness. A6 involves randomness but not with numbers.

A5 is UNFAIR since no-one writes down numbers at random; there is usually a preponderance of 3's and 7's, and rather fewer 1's and 9's. A6 should be FAIR if the pieces of paper are identical, folded the same way and there is adequate mixing. This may need stressing with the pupils.

A7
This is a summary of results so far. Question b indicates that even in a fair game random variation is present. With 20 counters it is not possible for three players each to win the same number of counters. Question c is designed to draw out the fact that playing the game a large number of times leads to greater certainty about its fairness or otherwise.

Question d is meant to direct attention to rolling a die or tossing a coin.

Section B

The equally likely outcomes are here formalized to generate random numbers.

B1
These four methods can be tried individually or in pairs, one member generating the numbers and the other recording. Most pupils need only two of the four methods to understand the ideas (Methods 1 and 4 take longer). All are fair methods.

Method 1
This describes the construction of a decagonal spinner. A professionally made one can be used or the teacher might-like to prepare sufficient in advance as 'home-made' ones take time.

Method 2
This is shown in Table 3 to be fair. Examples should be emphasized.

Method 3
This involves using an icosahedral die. These can be bought from E.J. Arnold or Technical Prototypes. It is possible to make an icosahedron but it is time-consuming. Ignore this method if no icosahedron is available.

Method 4
(Pupils familiar with binary numbers may be interested to know that this method is based on them, putting 0 for T and I for H. THHH becomes 0111, which is 7 in binary).

The use of Table 4 should be considered with the three lines of Table 5 before the experiment is done.

In each of these methods the 40 numbers should be recorded as a list and on the tables on pages Rl and R2.

The bar charts showing frequency distribution are optional. They can be expected to be uniform, and the effect of random variation can be lessened by collecting class results.

Note that non-random numbers can also lead to a uniform distribution. E.g. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 is uniform but not random. The unique feature of random numbers is that at each step each number is equally likely to occur.

B2

1. This should be more or less random.
2. This will not be random since there can be no 0's and smaller numbers are more likely.
3. This will be more or less random unless you choose a page on which a single firm has a lot of telephone numbers in sequence.
4. This is an important revision question similar to A5. There is usually a dearth of 0's and 9's and an absence of two successive occurrences of the same digit.

B3
Random number tables are introduced as the printed results of carrying out some random experiment. Usually computers use arithmetic procedures to produce lists of numbers which satisfy all the standard tests for random riumbers.

Other uses of random number tables are possible and could be shown to more able pupils. For example, to get random numbers from 1 to 17 you could use two-figure random numbers. If the tens figure is even, call it 0; if the tens figure is odd, call it 1; and ignore 00, 18 and 19.

So

77 04 01 09 73 59 84

becomes

17 4 1 9 13 19 4

(The abundance of 7's and 9's in the truly random numbers of Table 6 should be pointed out as being quite possible.)

*B4
This is an optional section for the more able/faster pupils to do while the slower ones are catching up.

Pupils can be numbered alphabetically or by numbering desks, among other ways. It is usual in this context to have sampling 'without replacement', so if the same random number occurs again it is ignored. This is the point being made in Questions b and c. It may be worth discussing this with the class if the problem arises. An example of sampling 'without replacement' is found in C1b.

Section C

Leaflets on premium bonds are obtainable from most post offices. (Children cannot buy premium bonds, but adults can hold them in a child's name.)

C1
Although premium bonds cost £1, they have to be bought in multiples of £5. This is not mentioned in the Pupil Unit since it would confuse the setting up of C2, the Ernie game.

1. It is fair.
2. It is fair since the ruling applies to all bonds.
   It is most unlikely since the proportion of winning bonds is so small. Allowing a bond to win two or more prizes is equivalent to sampling with replacement.
3. Choice should not make any difference, as is illustrated by the game.
   

C2
As described the teacher can take the part of the post office (or if you wish, this task can be given to a pupil). A 10 by 10 grid is useful here (see Equipment and Planning). As described the class will have to be split into 10 groups. If you wish, you can split the class into groups of size 11 and have several games being played at once, with each pupil buying bonds individually and one of the group acting as the post office. The suggestion in the Pupil Unit that groups should discuss strategies of choosing bond numbers is to encourage the airing of some fallacies connected with such things as lucky numbers and superstitions.

If you play the game as described, the use of a 10 by 10 grid is to ensure no two groups buy the same numbered bonds. Alternatively you could use a set of tickets numbered 0 to 99, and the groups could buy their 10 tickets in turn from you. (Books of raffle tickets can be used.) Any simple fair method of choosing an order can be used.

In order to make the prizes sufficiently large some compromise had to be made in matching this game with reality. The interest rate assumed is 10% per month to give a monthly draw. Alternatively you could use the same figures with a 10% per annum prize money and call each game a yearly draw.

C3
A frequency table or bar chart showing the number of groups winning different amounts of prize money over the 10 months might be illuminating, especially to see how many groups won over Z10 and by how much.

In Question d pupils should realise that numbers successful in the short run have the same chance as any other set of numbers in the long run. Question e should show that pupils with the greatest choice of bond numbers had no better chance of winning, and so it does not matter which numbers you choose. In Question g the method is fair by definition of random numbers.

It may be useful to pool all these ideas together in a final class discussion. Although the money used to purchase premium bonds can be recovered, this does not mean that there is no loss involved. There is a loss of purchasing power of the £1 due to inflation. Also, if otherwise invested at 10% (simple) interest, all groups in the Ernie game winning less than £10 can be considered as losers. If the current rates of interest are higher than the effective rate being given out as prizes for premium bonds, then actual bond holders could be considered losers. Brighter pupils might like to see the effect of, say, a 15% rate being available elsewhere. People may nevertheless still buy premium bonds because of the small chance of winning a very high prize.

C4
These questions can be answered using the complete results recorded during the Ernie game within the same sort of random variation shown in C3. It should be brought out that no strategy is better than any other.

Answers

A1	c	Fair
A2	c	Unfair
A3	c	Unfair
A4	c	Unfair
A5	c	Unfair. See detailed notes.
A6	c	Fair. See detailed notes.
A7	a	See above.
	b	No. See detailed notes.
	c	See detailed notes.
	d	See detailed notes.
B2		See detailed notes.
B3	c	0,4,0,1,0,3,4,3,5,5
	d	4,3,4,3,5,5,6,1,2,3
	f	4, 1, 9, 73, 35, 12, 39, 43, 64, 40 are the most likely answers.
B4		See detailed notes.
C1		See detailed notes.
C3	d	No
	f	No, not in the long run.
	g	Yes

Test Questions

(When there is a True/ False at the end of the sentence, you are to say if the statement is true or false.)

1. Ann, Brian and Charles play some games 21 times. Here are the results:
   
   

   Game 1 Number of wins Ann 6 Brian 8 Charles 7

   Game 2 Number of wins Ann 4 Brian 12 Charles 5

   Game 3 Number of wins Ann 5 Brian 9 Charles 7

   1. In a fair game they will always all win exactly seven times. (True/False)
   2. Game I is probably a fair game. (True/False)
   3. Game 2 is probably a fair game. (True/False)
   4. What would you do to help decide if Game 3 is fair or unfair?
2. In ten single-figure random numbers every figure from 0 to 9 will always be there once. (True/False)
3. Describe one way of obtaining random numbers using coins, dice or spinners.
4. Ann is obtaining random numbers. She writes down: 3, 2, 1, 7, 3, 8.
   1. The next number cannot be 8. (True/False)
   2. The next number is more likely to be 5 than 3. (True/False)
5. Numbers written from your head are random numbers. (True/False)
6. Here are some ways of obtaining numbers. Say whether the numbers are random or not random.
   1. Write down the size of shoe worn by pupils in your class. (Random/Not Random)
   2. Count the number of words on a page in a book. Write down the last digit (units digit). (Random/Not Random)
   3. Measure the height of pupils in centimetres. Write down the first figure. (Random/Not Random)
7. Here is a line from a table of random numbers.
   87 71 56 03 85 03 11 69 23 98 78 64 52 19 04 39
   Starting from the beginning of the line each time, use it to write down five random numbers.
   1. from 0 to 9
   2. from 3 to 7
   3. from 0 to 74
   4. from 25 to 70
8. If you buy a premium bond, you can't get your money back. (True/False)
9. In the Ernie game:
   1. Odd numbers are more likely to win than even numbers. (True/False)
   2. Some people win more prize money than others. (True/ False)
   3. Small numbers always win more prizes than large numbers. (True/ False)
   4. All sets of ten numbers have the same chance of winning. (True/ False)

Answers

1	a	False
	b	True
	c	False
	d	Play it more times.
2		False
3		Any of Methods 1-4, Section B1.
4	a	False
	b	False
5		False
6	a	Not random
	b	Random
	c	Not random
7	a	8, 7, 7, 1, 5
	b	7, 7, 5, 6, 3
	c	71, 56, 3, 3, 11
	d	56,69,64,52,39
8		False
9	a	False
	b	True
	c	False
	d	True

Connections wlth Other Published Units from the Project

Other Units at the Same Level (Level 1)

Shaking a Six
Practice makes Perfect
Tidy Tables
Wheels and Meals
Probability Games
If at first...
Leisure for Pleasure

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

Level 3

Cutting it Fine
Pupil Poll

Level 4

Choice or Chance
Testing Testing

This unit is particularly relevant to: Mathematics.

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 teaching the 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

Number	Tally	Total
0
1
2
3
4
5
6
7
8
9

Table 8 - Tally chart for first method Chosen method.......

Table 9 - Tally chart for first method Chosen method......

Table 10 - Numbers using Method 4

Page R2

Number	Tally	Total
0
1
2
3
4
5
6
7
8
9

Table 11 - Tally chart for Method 4, tossing four coins.

Table 12 - Tally chart for random number tables.

Score sheet for Ernie game

GROUP NUMBER............... BOND NUMBERS

Prizes	Value	Month 1	Month 2	Month 3	Month 4	Month 5	Month 6	Month 7	Month 8	Month 9	Month 10
1	10p
2	10p
3	10p
4	10p
5	10p
6	10p
7	10p
8	10p
9	10p
10	10p
11	50p
12	50p
13	50p
14	50p
15	£1
16	£1
17	£5
Amount Won

Total amount won after 10 months £........

Page R3

Random Numbers

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