developing a feel for probability

Developing a Feel for ProbabilityAuthor(s): Richard EnglishSource: Mathematics in School, Vol. 21, No. 2 (Mar., 1992), pp. 14-18Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214850 .

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)eveloi a eel

FOR

)ro by Richard English Mathematics Advisory Teacher, Humberside

Traditionally pupils in secondary schools have not done any work on probability until at least the third year (now year 9). To confirm this, one only has to look at the popular maths schemes and texts that have been used in secondary schools over the last ten years or so, many of which are available and still being used today. The relevant chapters in such books usually follow a fairly standard format; a swift introduction to a probability scale ranging from 0 to 1; instruction in calculating probabilities using fractions and/or decimals; and very soon after that pupils are expected to multiply and add these probabilities. This rapid progress is slowed in some texts by tossing coins, rolling dice and flicking drawing pins in the early stages but these activities are quickly followed by the number- crunching.

Probability work has always been left until the latter secondary school years presumably because the approach adopted required pupils to be competent in their use and understanding of fractions and decimals. But as well as assuming this arithmetical competence, the schemes and. texts make an even bigger assumption; that the pupils have an understanding of the concept of probability. Progress through the work, from first introductions through to the combining of tricky theoretical probabilities, has often been far too rapid for pupils to develop any real understanding of what probability is all about.

With the introduction of the National Curriculum hope- fully this situation is changing and pupils should now be experiencing probability at a much earlier age. It is through these early experiences that they can develop their own

understanding of probability, thus making the numerical and arithmetical aspects that are introduced later (at about level 5) more meaningful.

The Maths Resource sheets here show four examples of activities that I have used with pupils at Key Stages 2 and 3 (a fifth activity was featured in "Probhex-a game about hexagonal probabilities", Mathematics in School, Vol 20, No 3). The aim of these activities is to assist in the development of an understanding of probability by considering the much used notion of fairness.

I usually start by having a raffle. Each pupil writes his or her name on a slip of paper and places it in the hat. Meanwhile I, in view of the whole class, write my name on several pieces of paper and attempt to place them in the same hat. Inevitably comes the cry "It's not fair!", which leads us into a very worthwhile discussion of what we mean by "fair". The pupils then set about the task of deciding whether or not the enclosed games, and activities are fair. I like the pupils to make a prediction first, supported by reasons, as this provides the basis of much discussion. The games and activities themselves raise many important points for discussion; how many times must the activity be done in order to reach a reliable conclusion, why is the game not fair, how can it be altered to make it fairer?

As well as providing valuable experience of probability for younger pupils these activities can also provide a challenge for older pupils right up to sixth form level. Calculating the theoretical probabilities of the various possible outcomes in some of these activities is no easy task and if you have any doubts about this then try working out the expected winnings per game in the Fairground Game and also read the article in Mathematics in School referred to above.

14 Mathematics in School, March 1992



Mathematics in School, March 1992 13

UPS

AND

DOWNS

BUT

IS IT A FAIR

GAME?

WIN

Place

counter

on

start.

Flick

a

coin

-

Heads,

move

forward

one

space

-

Tails,

move

forward

two

spaces.

Watch

for

the

arrows.

Keep

flicking

until

you

win

or

lose.

START

LOSE



Fairground Game

WIN

2p WIN

5p

but is it fair?

WIN

o10p WIN

20p

WIN

30p

WIN

40p

WIN

50p

WIN

5p WIN

10op /WIN

20p

WIN

30p

WIN

u

40p

WIN

50p

START

Place a 10p piece on the start (non-returnable)

Roll 2 dice (6 sided) Move ONE SPACE

according to total

Repeat until you win.

6, 7, 8

2, 3, 4, 5 9, 10, O11, 12




Mathematics in School, March 1992 13

LEAPFROG

Flick

a

coin

four

times.

Move

one

space

at

a

time.

HEADS

TAILS

Where

will

I

be

after

four

flicks?

L

E

A

P

F

R

O

G




Roll

two

dice

Work

out

difference

0,

1

or

2

move

ONE

space

this

way.

3,

4

or

5

move

ONE

space

this

way.

Repeat

until

you

win

or

lose.

WIN

I;

ILOSE



developing a feel for probability

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