Brief Description

The central theme is that of using simulation to model real-life situations. Children often collect sets of cards; the first simulation investigates how long it would take to collect a set of four. Other simulations use dice and random numbers to model booking seats on a minibus, finding the right key for a door, the weather and being stopped at traffic lights.

Design time: 5 hours (B4 and C3 optional)

Aims and Objectives

On completion of this unit pupils should be able to set up, and improve if necessary, a simulation using (i) playing cards, (ii) a die, and (iii) a random number table; to model simple real-life situations.

They will have practised the use of a random number table to select series of random numbers to suit varying probability requirements.

They meet examples of everyday situations which can be rephced (to various degrees of accuracy) by simple probability models and simulated in the classroom.

They should be more aware of the advantages and possible disadvantages of replacing real-life situations by models.

Prerequisites

Pupils should be familiar with simple fractions and proportions, and should be able to mrry out division to one decimal place.

Equipment and Planning

Dice and packs of playing cards are required for the class, ideally one for each pair of pupils.

The record pages R1 to R3 contain blank tables and answers to be completed. Page R4 is a set of random number tables.

Concepts in probability are rather difficult; consequently the success of this unit depends greatly on the level of participation of the teacher. It is not recommended that this material be treated wholly as individually-based learning.

Section A is intended for all pupils and develops the idea of a simulation as an experiment which models the essential features of the real-life situation. Two problems are investigated: the collection of sets of cards, or the like, and the operation of a minibus service.

Section B introduces random numbers and random number tables. B4 is optional, designed for more able pupils, and develops the use of random numbers beyond the general demands of the unit.

Section C provides further practice in the use of random numbers by modifying the minibus example. It introduces two further, more sophisticated situations (one optional in C3, again for more able pupils): key selection and the passage through two sets of traffic lights.

Detailed Notes

When probability is used to model real-life situations, the calculations can become quite complex. This is apparent from the theoretical solutions presented later.

This unit helps pupils build up an intuition for such ideas. It shows how real-life events can be likened to random processes by assigning and combining the relevant probabilities. The rules of probability are thus being built on intuitive foundations.

Monte Carlo methods are used - this means using random numbers for simulating chance events. A single trial is unpredictable, but repeated trials indicate the distribution of results that might be expected. This requires many results. It is suggested that class results be combined, though it is essential that all pupils do some simulations individually. They then develop a feeling for probability by seeing random events in different settings.

As described in Section A below, it is better to start with random number generators: spinning discs, tossing coins, throwing dice, etc. Class results can be used to illustrate that all digits, pairs of digits, and so on appear independently and with equal frequency in the long run.

These features of random numbers are crucial and could be emphasized by using biased spinners. They are also considered in the Level One unit Being Fair to Ernie.

The object is to develop a feel for random numbers. The simulations in this unit use randomizing devices to model ideal problems.

Teachers desiring to find out more about theoretical aspects of simulation could consult the following books.

1. Monte Carlo Methods, by J. D. Hammersley and D. C. Handscomb, (Methuen, London, 1964.) This deals with some of the theory.
2. The Teaching of Probability and Statistics, A. Engel in L. Rade (ed) (Almqvist and Wiksell, Stockholm, 1970; distributed in this country by John Wiley). This looks at some simulations that can be done by schoolchildren.

Section A

We suggest that you introduce this section with a class demonstration and discussion. Throw a die, call Heads if an even number results, call Tails when odd. Repeat this several times, then ask the pupils to guess the rule. Alternatively, call Boy for odd and Girl for even to simulate births by sex.

This can lead to the consideration of the possible advantages and disadvantages of simulations: speed, convenience, danger of over-simplification. One can then show that many aspects of the real world can be considered as chance events that can be modelled by the appropriate use of random numbers, for example, male/female birth, car/van passing, rain/dry.

Thus instead of a real process, it is sufficient to show the output of a randomizing device, even though equally likely probabilities may not be involved. This is the essence of Monte Carlo methods.

Two particular simulations are suggested in the pupil notes for discussion. Tossing a coin four times can simulate the number of boys and girls in a family, using Heads for girls and Tails for boys. Book cricket is a game played by following a text and using a = maiden ball, b = bowled, c = caught, d = 2 runs, e = 1 run, f = 4, g = 6, etc.

Pupils may well have their own rules for this game, which is not a particularly realistic model of a game of cricket, since little attempt has been made to match up the probabilities.

Pupils may benefit from a demonstration of A1 and A2 from the teacher. If resources are short, A1 and A2 could be attempted by different parts of the class. A whole pack of cards is not essential, though equal numbers of each suit are; the replacement of cards is necessary if there are only a few cards, but this is difficult to organize.

A1
The first example should be familiar to pupils, and the teacher may find current examples a useful aid.

Questions a, b and c are intended for discussion and could be extended further if required to d 6 cards and e 12 cards.

The pupils may require some help in the execution of the experiment. Teachers may consider the use of a flow chart to clarify instructions. A blank table, Table 2, on page Rl is provided for recording the results. Collect the class results for pupils to use in A2.

A2
The cards involve selection without replacement, and so the probabilities do change after each card is drawn. This makes it slightly quicker to collect a set (the mean is reduced by about 0.3). It should take, on average, about eight cards to complete a set, but the distribution is rather skewed. c is rather difficult theoretically.

Class results could be collected with a table running from 4 (the least necessary) to about 25, though theoretically 40 might be required. If there is time, a bar chart could be drawn to illustrate the skewed distribution.

A3
The minibus problem is a standard one with bookings: it is binomial with n = 12 and p = ¹/₆. An owner would probably not overbook, but with present economic stringencies it might make the difference between profit and loss. However, overbooking can result in disgruntled customers who are turned away. Certainly airlines have overbooked and been taken to court for it.

Here dice are used to simulate chance, and pupils may need extra guidance on how to perform the experiment and fill the table. Table 3 on page Rl is provided for recording individual results. Class results should be collected and discussed, as five is rather a small sample size.

Question c is a comment on individual experiments and may be taken up as part of the class discussions arising from d and e.

It is possible that an unacceptable number of people may be turned away. The equally likely hypothesis is tenuous. People may come without booking. Demand may vary.

Section B

Random numbers are introduced in this section. Pupils may experience difficulties and will certainly benefit from class discussion, with considerable teacher involvement.

It may be useful to try to show children that numbers called out by them 'at random' are not really random. Ask them all to write down a number 'at random' choosing from the numbers 0 to 9. Collect the results. Usually there are fewer zeros than would be expected from genuine random numbers.

A point worth stressing is how to get a probability of 1 in 6, by ignoring 7,8,9 and 0. One could introduce modulo method (as outlined in B2 for a chance of 1 in 2) but this is left to the discretion of the teacher.

Brighter pupils may appreciate trying to make more efficient use of the random number tables by minimizing the number of digits that have to be discarded. Thus a probability of l in 3 is the same as 3 in 9, and all digits 1 to 9 can be used with only the zero being discarded. 4 out of 7 is the same as 56 out of 98, and so a more efficient use of two-figure random numbers can be made, and so on.

The use of random numbers is also covered in the Level One unit Being Fair to Ernie.

B1
If time is available, pupils could usefully construct and use their own table of random numbers from dice throws.

B2
Page R2 provides copies of the extract of the random number table used in illustrations and can be used as a work-sheet.

You might like to consider the introduction of the following method for a chance of 1 in 2:

If the number is 0,2,4,6 or 8, the event takes place.
If the number is 1,3,5,7 or 9, the event does not take place.

B3
This is a further development of the ideas in B2. The results obtained in B2a, b, and c, may be helpful in B3a, c and d.

You might consider using local weather data to relate to local events. For example how many school cricket matches are likely to be rained off this season, based on the proportion of days when there is sufficient rain to cause this. Another possibility is to consider how many times a week you are likely to have to water an outside plant which requires water or rain every other day.

The data used to introduce this section was obtained from Sheffield Weather Summary 1976 (Sheffield City Museums).

The random numbers model is quite crude, since it implies independence of weather from day to day.

If 3 in 5 is written as 6 in 10, the simulation can use all the random numbers with:

If the number is 1,2,3,4,5 or 6, it rains.
If the number is 7,8,9 or 0, it does not rain.

*B4
This section is optional and intended for more able pupils. It applies the ideas already introduced to probabilities requiring the selection of two-digit random numbers.

You may consider refining methods for b 7 in 30, and d 1 in 13, to make more efficient use of the random number table.

Section C

In this section we return to the minibus problem and introduce other situations which can be simulated with the help of the random number table provided with this unit.

C1
Another piece of information is introduced in the minibus problem. This is designed mainly to afford pupils practice in the use of random numbers without making them consider something new.

Table 4 on page R1 is provided for recording individual results. Again, it would be helpful to compile class results before the discussion. In the discussion, you may wish to include:

5. Have any important features been ignored?
   How might they be taken into account?

C2
This situation may be introduced by demonstrating with, say, three similar keys for a classroom or cupboard door. The three ways of choosing should confirm what one could guess beforehand yet it is still worthwhile to show how one can use the simulations. The advantage of the simulations is that it gives some measure of the difference in efficiency between the various methods. This would take a long time experimentally. If time is short, only one simulation need be done, but it is desirable to use several to show how you can adapt the basic model to fit the different situations.

Tables 5, 6 and 7 on page R3 are provided for the recording of the results of a, b and c respectively.

C3
This section is optional and is intended for more able pupils. Class results could be compiled before answering e. The traffic light example can be illustrated by a tree diagram:

If built-up by pupils or on the blackboard, red and green lines may enhance this diagram.

The experimental proportions will not be identical to the theoretical probabilities, but the multiplication rule will still hold. This can be used with more able pupils to help lay the foundations of multiplication of probabilities.

Answers

A1	a	See detailed notes.
	b	See detailed notes.
	c	See detailed notes.
A2	a	The mean should be about 8.
	b	4 and 40 are the minimum and maximum possible values.
	f	See detailed notes.
	g	See detailed notes.
A3	b	Using binomial with n = 12, p = 1/6 theoretical answers for five trials are: 1.35, 0.56, 1.48, 1.61.
	c	See detailed notes.
	d	See detailed notes.
	e	See detailed notes.
B1	a	5 1 9 3 7
B2		Many possibilities, e.g.
	a	1 the person does not come 2,3,4,5, the person does come 6,7,8,9,0, ignore
	b	Use 1; 2,3,4,5,6,7,8; 9,0; as the three categories.
	c	Use 1; 2 to 9 and 0; as the first two categories.
B3		Many possibilities e.g.
	a	Use 1,2; 3,4,5; 6,7,8,9,0; as the three categories.
	b	Use 1,2,3,4; 5,6,7; 8,9,0; as the three categories.
	c	Use 1,2,3; 4,5,6,7,8; 9,0; as the three categories.
	d	Use 1,2,3; 4,5,6,7,8,9,0; as the first two categories.
C1		Results will vary.
C2	a	The mean should be about 6.
		The mean should be about 51/6.
		The mean should be about 31/3.
C3	a	3/5.
	b	1/3.
	f	The exact probabilities are 4/15, 8/15, 1/5

Test Questlons

1. By giving an example explain what is a simulation.
2. Write down two advantages and one disadvantage of a simulation.
3. In a list of random numbers, what is the chance that the next number will be:
   1. 7
   2. An even number
4. The chance that Fred wakes up late is 2 in 7.
   We can simulate this using random numbers:
   If the number is 1, 2 Fred wakes up late.
   If the number if 3,4,5,6,7 Fred does not wake up late.
   From the list of random numbers given below:
   1. Make out a list of the random numbers you would need for the experiment.
   2. Underline each number in your list which means that Fred is late.
      5 2 1 4 8 9 5 6 6 5 9 4 1 3 0 8 0 3 1 8
      1 0 0 4 2 3 9 4 3 5 6 3 4 2 8 6 6 7 2 6
5. Write down a rule for simulating a chance of
   1. 1 in 8
   2. 2 in 5
   3. 4 in 9
6. The Sheffield Weather Summary 1976 shows that there were six rainy days in April 1976. Describe a simulation to find out how many rainy days there would be in one week.

Answers

1. An experiment which uses a randomizing device to model another situation.
2. Advantages: quick and easy
   Disadvantages: may not be a very good model
3.  
   1. ¹/₁₀
   2. ¹/₂
4.  
   1. 5 2 1 4 5 6 6 5 4 1 3 3 1
      1 4 2 3 4 3 5 6 3 4 2 6 6 2 6
   2. 5 2 1 4 5 6 6 5 4 1 3 3 1
      1 4 2 3 4 3 5 6 3 4 2 6 6 2 6
5. e.g.
   1. 1; 2,3,4,5,6,7,8; 9,0; are the three categories.
   2. 1,2; 3,4,5; 6,7,8,9,0; are the three categories.
      or 1,2,3,4; 5,6,7,8,9,0; are the first two categories.
   3. 1,2,3,4; 5,6,7,8,9; 0; are the three categories.
6. P(rain) = ⁶/₃₀ = ¹/₅ = ²/₁₀
   Use 1,2 for a rainy day,
   3,4,5,6,7,8,9,0 for a non-rainy day.
   Read off seven random digits to find out the simulated number of rainy days.

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 1)

Shaking a Six Being
Fair to Ernie
Wheels and Meals
Probability games
Practice makes Perfect
Leisure for Pleasure
Tidy Tables

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

Level 2

On the Ball
Seeing is Believing
Fair Play
Getting it Right

Level 3

Car Careers
Net Catch
Cutting it Fine
Multiplying People

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 number in the left-hand column refers 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

Trial	Guess	Number of cards used
1
2
3
4
5
6
7
8
9
10
Totals

Table 2

Table 3

Table 4

Page R2

a	For a probability of 1 in 5: If the number is   the person does not come. If the number is   the person does come. If the number is   we go to the next number. 9 1 9 3 8 8 5 6 3 5 7 6 9 7 3 5 1 9 3 7 1 4 6 6 0 7 4 6 5 0 5 8 0 8 7 3 4 2 9 7 2 0 4 2 6 4 6 8 0 0 So a list we could use is
b	For a probability of 1 in 8: If the number is   the person does not come. If the number is   the person does come. If the number is   we go to the next number. 9 1 9 3 8 8 5 6 3 5 7 6 9 7 3 5 1 9 3 7 1 4 6 6 0 7 4 6 5 0 5 8 0 8 7 3 4 2 9 7 2 0 4 2 6 4 6 8 0 0 So a list we could use is
c	For a probability of 1 in 10: If the number is   the person does not come. If the number is   the person does come. If the number is   we go to the next number. 9 1 9 3 8 8 5 6 3 5 7 6 9 7 3 5 1 9 3 7 1 4 6 6 0 7 4 6 5 0 5 8 0 8 7 3 4 2 9 7 2 0 4 2 6 4 6 8 0 0 So a list we could use is

Page R3

Trial	Key	Number of keys tried
1
2
3
4
5

Table 5

Table 6

Table 7

Table 8

Page R4

Random Numbers

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