Brief Description

Basic ideas of probability are introduced through games using dice and coins. The games develop in complexity, demanding simple strategies. The main part of the unit concludes by encouraging pupils to compare relatively complicated probabilities without detailed calculations. There is a final optional section on colouring shapes which introduces basic ideas of permutations.

Design Time: 4 hours (Sections A7, A9, B2, B3 and C are optional.)

Aims and Objectives

On completion of this unit pupils should be able to understand probability in terms of allocating particular numbers between 0 and 1 for simple equally likely cases (linking these to long-run expectations), also as a development of the ordinal scale from impossible through likely, to certainty.

They will have practised the collation, tallying, tabulating, graphing and analysis of results, and the reading and interpretation of data presented in tabular and graphical form.

They will be more aware of the nature of variation and probability, especially in the context of experiments with coins and dice. They should also be more aware of phenomena which exhibit random variation.

Prerequisites

Pupils should have met simple tallying and bar charts from discrete data and be able to order numbers to one decimal place between 0 and 1 and be able to write down simple fractions.

Equipment and Planning

About a dozen dice and 300 counters will be needed. An optional section requires spinners and non-cubical dice. Pages Rl and R2 are available for recording the results of Section A. Pupils could work in pairs or individually as required. Occasional class discussions may be beneficial.

Section A begins with the examination of a game based on the throwing of a die. The question of fairness and bias precedes the more formal introduction of probability. Sections A5 and A6 extend the idea of probability to more subjective topics. A7 is optional, providing the opportunity for faster pupils to investigate non-standard dice or spinners. A8 introduces a game with two dice, which is adapted in the optional A9.

Section B introduces a game requiring the judgement of complex probabilities. Experimental methods are used in their evaluation. B2 and B3 are optional, providing two further games using dice and coins.

Section C is intended for relaxation, although it does lay down some combinatoric ideas. These ideas are not pursued to any great depth in the pupil unit but can be developed if required.

Detailed Notes

Section A

A1
Teachers may well feel that a class demonstration of the game would be a desirable opening.

Although Alan and Brian can both gain counters in three ways, thegame is not fair - on any throw it favours Brian in the proportion 12:9. Pupils should find it instructive to discover this by playing the game five times and pooling the results. Page Rl is used to record the results of these 10 games.

Collecting the class results of f and g should give a clearer picture and lead to a general discussion.

One way of making the game fair is to allow only one counter to be won when the appropriate score shows. Another alternative is to let Alan win on 2, 3 and 5 and Brian on 4 and 6. It is not easy to work out the probability of B winning. In fact it is 0.663. The mean number of throws before the game is completed is a little over 7 (the mode is 3 and the median is 5).

Results from trial schools gave A 414 wins to B's 685.

One suggested method of class organisation is to arrange for each pair to have Odds on the left and Evens on the right. On the completion of the games, ask the more successful member of each pair to stand. You can then quickly distinguish between Odds and Evens.

A2
It is valuable to collect the class results and use them to construct a larger version of Figure 1 to reinforce the concept of settling-down suggested in c. The bars should be approximately equal and even more so for a class bar chart. The appropriate column would be taller if the die were loaded.

A3
This section reinforces the work already done and links in the term unbiased with probability.

A4
This section is designed to bring out the one main aspect of random variation that there should be some differences between the frequencies, but not too much. The results most likely to be authentic are in column (ii). There is no variability in (i); (iii) has a large number of twos; while (iv) has too much variation.

A5
This section may well provoke some discussion and disagreement. It requires the use of familiar words to describe future events. There is a combination of subjective and objective probability based to a greater or lesser extent on statistical evidence.

It is expected that past experience will be used to give an assessment of likelihood.

The idea of odds, as in betting, could be introduced here.

A suggested order of terms is: impossible, highly improbable, unlikely, fifty-fifty, probable, very likely, certain; although people by no means agree on the order of such phrases. It is thus desirable to move to a numerical scale, as is done in the next section.

A6
Pupils work out simple probabilities; some may require extra help. Some familiarity with decimal numbers is required to forge the link with A5. The number line, from 0 to 1, could be drawn (perhaps on an overhead projector) and markers used to position the statements from A5c.

*A7
In making fair spinners, it is easier to spin arrows than home-made spinners. One can push a drawing pin through the centre of a circle drawn on a piece of card. Through the pin-point put a thin pointer which can be spun freely while the card is held firmly. The circle can be painted in the desired ratio. Alternatively, the circle can be spun and the arrow drawn outside the card.

The experiments should consist of a large number of spins or throws, followed by analysis of the frequency of each possible outcome, and should be related back to the number of faces or the proportion of the spinner giving a success.

Some possible spinners are shown below.

A8
This section examines a game which uses two dice.

Brighter pupils should be able to see theoretical differences when shown the table of the 36 equally likely outcomes see Table Tl. This should be done after the game has been played.

Table T1 - The sum of two dice

Initially the game may seem to favour A since he has seven winning totals to B's four. In fact the game favours B since he has 20 of the 36 equally likely outcomes to A's 16. The probability that 8 gets five counters before A is 0.635.

*A9
This optional section contains a variation of the game in A8. Again, the two-way classification table may prove helpful and is shown in Table T2. Here A has the better odds at each throw (23:13) and has a probability of 0.81 of winning the game.

Table T2 - Multiplying two dice.

Section B

The obstacle race involves complex probabilities. However, the choices are fairly clear-cut. Teachers may wish to add further obstacles of a similar nature, maybe using spinners, biased dice, etc.

B1
This section introduces the ideas of the game and leads pupils to methods of choice between obstacles, after which the complete game is presented. Pupils could work in teams of four, with each pair testing two sets of obstacles. The experiments of A8 should give a good indication of the best route. It could be played as a class race, with the obstacles drawn on an overhead projector, and all the chance events performed simultaneously. The best route is 1b, 2a, 3b, 4b. The odds for each obstacle, taken in order for a:b, are 1:2, 5:3, 3:4 and 5:9.

*B2
This optional section introduces an American (or rugby) football game; in fact, it is similar to the game in A1. Team A has the slight advantage. It may be used as reinforcement if desired.

*B3
Again optional, this section provides a game based on tennis and can be made more realistic by using different chance methods of deciding when a shot is good (e.g. higher probability of putting second serve in, lower probability of returning a good first serve). The idea of simulation is developed in the Level One unit If at first ... As described, this game is, surprisingly, fair. As an alternative to using a coin, a die can be thrown with Odd and Even used for Head and Tail.

Section C

This section is optional and is intended to introduce simple ideas of combinations and permutations. It is essential in this work that pupils develop a logical strategy to ensure that all possible combinations are tried once and only once. It is put in a practical context to encourage pupils to develop such strategies. Later these results can be formalized in tree diagrams and formulae. Coloured rods or unifix cubes can be used to make different coloured towers or investigate the number of ways of colouring tricolour flags:

Pupils may begin to see the number of patterns if you show them an uncompleted table of their results from John's boat, as in Table T3. The numbers in italics can be filled in by the pupils.

Table T3 - Number of ways of colouring with no repeats.

Generally, for n colours and r parts:
If colours are not to be repeated, there are:
If colours can be repeated, there are n^r ways.

Answers

A1	a	3
	b	3
	c	See detailed notes.
	h	See detailed notes.
A2		See detailed notes.
A3	b	Probability of each face is 1/6
A4		See detailed notes.
A6	b	1/3
	c	1/2
	d	1/3
	e	5/6
	f	1/2
	g	1/3
	h	0
	i	1
	k	Impossible, Certain
A8	a to c	See detailed notes.
	g	Theoretical answers are 7, 2 and 12, but the actual answers depend on the results of the dice throwing.
	h	See detailed notes.
B		See detailed notes.
C	a	6
	b	6
	d	12
	e	24
	f	4
	g	16

Test Questions

1. If you throw an unbiased die, what is the probability of getting:
   1. a three,
   2. an even number,
   3. an eight?
2. A strange die has six faces labelled 1, 1, 2, 3, 4, 5 and no six. Draw the type of bar chart you would expect if you threw the die 60 times.
3. Alan, Brian and Carol each rolled a fair die 240 times. The table shows the results they told their teacher. One of them was telling the truth, the other two were lying.
   

   Number	Frequency
   	Alan	Brian	Carol
   	24	40	37
   	26	40	42
   	33	40	36
   	27	40	48
   	96	40	36
   	34	40	41

   1. Who told the truth?
   2. Why do you think it was the other two who were lying?
4.  
   1. Write down an event which is certain to be true.
   2. Write down an event which is unlikely to be true.
   3. Write down an event which is very likely to be true.
   4. Write down an event which is impossible to be true.
   5. Suggest a decimal number for the probability for each of the events a to d.
5. You have to choose between the following two obstacles:
   1. Throw two fair dice and get a total score of six,
   2. Spin arrows on these two spinners so that both arrows point to the unshaded section.
      

      a How would you decide which is the easier obstacle?

Answers

1	a	1/6
	b	1/2
	c	0
2		A bar chart with heights about 20, 10, 10, 10, 10 is better than one showing them exactly with these frequencies.
3	a	Carol
	b	Alan gave too many fives. Brian did not give any variation.
4	a to d	Accept any reasonable answer.
	e	1; something between 0 and 0.5; something between 0.5 and 1.0.
5		Carry out both experiments a large number of times and see which succeeded most often (or compare the relative frequency of the second with the 5/36 probability of the first or compare 5/36 with 1/8).

Connections with Other Published Units from the Project

Other Units at the Same Level (Level 1)

Shaking a Six
Being Fair to Ernie
Practice makes Perfect
If at first ...
Tidy Tables
Wheels and Meals
Leisure for Pleasure

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

Level 2

On the Ball
Fair Play

Level 3

Cutting it Fine

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 lntroduced 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

Game 1

Game 2

Game 3

Game 4

Game 5

Table 5 - Results of odds and evens

Page R2

	Tally	Number of times

Table 6 - The number of times each face of the die showed

Table 7 - Throwing two dice. The sum of two faces.

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