The Further Mathematics Support Programme
Making decisions using mathematics
This set of resources is designed to support teachers in:
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demonstrating ways in which GCSE Mathematics techniques can be used in reallife contexts;
illustrating the breadth of topics covered in A level Mathematics and Further
Mathematics by providing a ‘taste’ of decision mathematics;
providing materials which can be used to enrich the curriculum in Years 11 or 12.
It is particularly hoped that these resources will appeal to female students in these year
groups, and encourage them to see how continuing with the study of mathematics at AS
and A level would be beneficial.
A recent literature review by the Institute of Education (IoE) on behalf of the FMSP
indicated that girls are more likely to study A level Mathematics in conjunction with nonscience subjects than boys, and so providing examples and advice in mathematics that
relate to its relationship with subjects such as Biology, Chemistry Business Studies,
Economics, Geography, Psychology and Sociology may increase participation in A-level
Mathematics by girls.
This booklet starts with a Teacher guide, suggesting how the materials might be used; this
is followed by:
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Expected Monetary Value (EMV) worksheet and answer sheet
An Introduction to Decision Trees handout
There is an accompanying document entitled ‘Decision trees group tasks and answer
sheets’ and a powerpoint presentation entitled ‘Making Decisions using Mathematics’,
which provides additional activities and some information about risk analysis as a career.
The overall time required is likely to be at least 100 minutes. The activities would be
suitable for students who have studied Higher tier GCSE and have a good understanding
of probability trees.
For further information and guidance about girls and mathematics, see
www.furthermaths.org.uk/encouraging-girls-maths
Teacher Guide
Time
Activity
Resources
10 mins
Slide 2: Introduce the idea of decision making using mathematics
rather than just human judgement.
PPT
What sort of real-life scenarios might this be helpful for? e.g.
business decision making, deciding whether to enter the national
lottery, deciding whether to play a particular game, analysing
chances of winning in sporting competitions, etc….
Slide 3: Some examples of ‘key questions’ that the techniques in
today’s lesson might answer. Invite students to identify aspects
of the given scenarios relating to ‘chance’.
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15 mins
PPT
Deepwater drilling rigs can cost around $420,000 per day –
is it worth building the rig? How long will the rig produce
oil? Is it worth the costs of drilling?
How many possible combinations are there in the national
lottery draw? What is the chance of winning? What is the
expected win?
Dice games – if you had to pay compete in a dice game
and win only if certain scores are obtained, how would you
know whether to play or not?
Investment – if you have the opportunity to invest in a
business partnership for a new business, what would you
need to find out more details about? What is the degree of
risk? What might you expect to gain or lose in the first
year?
The organiser of a sports competition might take out
insurance against poor weather which prevents play – this
is called pluvius insurance or abandonment insurance
Slide 4: This is an aside to the main topic of the lesson, but it is
an interesting and relevant aspect to look at briefly, if time allows.
Introduce students to some basic investigation of the lottery
scenario e.g. If there are 4 balls labelled 1 – 4, how many ways
could two balls be chosen? What is there were 5 balls and 2
were chosen? Working this out individually is very time
consuming for picking 6 balls from 49. Introduce the nCr button
and how this ties in to the investigation. Now students can find
that the number of combinations is 49C6 = 13, 983, 816 and so the
chance of winning is around 1 in 14 million.
PPT
Calculators
Time
Activity
Resources
15 mins
Distribute copies of the EMV handout (page 4 of this booklet)
EMV
Handout
Slides 5 and 6: Introduce the idea of calculating expected
monetary value (EMV). Discuss how EMV can give a value for a
‘risk’ (financial loss) or for a ‘reward’(financial gain).
PPT
Ask students to work in pairs or small groups to consider the
questions on the EMV handout. After a few minutes of
considering the questions, review as a group:
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20 mins
Which of these outcomes are risks and which are rewards?
Which has the highest expected monetary value?
Which has the lowest expected monetary value?
If all of the outcomes listed occur, would they represent a
total loss or a total gain?
Answers are provided on an EMV handout including answers
(see page 5 of this booklet).
EMV
Handout
Answers
Distribute copies of the Introduction to Decision Trees handout
(page 6 of this booklet).
Introduction
to Decision
Trees
handout
Slides 7 - 11 : Explain the symbols used in decision trees and
sketch on the board how these would be positioned on a tree.
Slides 8 and 9 show a simple example of how to use a decision
tree – work through this example to illustrate the way a tree is
constructed and state the need to work backwards to do
calculations.
Now introduce the Dice Game outlined on slide 10 and encourage
students to sketch a decision tree, perhaps on a mini whiteboard
so amends can be made. Depending on the group, either allow
some time for pairs of students to draft their own decision tree, or
for students who are less secure with the ideas, go through the
production of the decision tree as a whole group.
When reviewing the calculations, discuss the importance of each
of the symbols. It is vital to work BACKWARDS when carrying
out calculations, which students might find unnatural as they will
be used to doing probability tree diagrams working from left to
right.
The answer is shown on page 7 of this booklet and is illustrated
step by step on slide 11.
PPT
Time
Activity
Resources
30
mins
or
longer
if time
allows
After reviewing the Dice Game, pairs of students could either pick
their own choice of task from the Decision Trees Groups Tasks
handout or tasks could be allocated by ability (Tasks C and D are
the most demanding).
Decision
Trees
Group
Tasks
Handout
Bring the group back together to discuss the tasks after they have
completed one (or more, depending on time available), or circulate
around the pairs of students to check answers.
If planning to discuss as a group, give students A3 paper to
produce their decision tree on so that they can be displayed to the
class.
10
mins
Slides 12 - 14: Finish the session give students an overview of the
role of risk analysis in business and as a career. Further
information can be found at:
http://www.prospects.ac.uk/financial_risk_analyst_job_description.
htm or http://www.theirm.org/
Close by signposting the FMSP website to students, where further
information about maths careers and the promotion of A level
Mathematics and Further Mathematics can be found.
PPT
The Further Mathematics Support Programme
Making decisions using mathematics
Expected Monetary Value (EMV)
An Expected Monetary Value (EMV) calculation is used to quantify the monetary risk or
reward of a particular outcome.
EMV = (probability of outcome) x (cost of outcome)
Consider the following risks and rewards associated with a large building project.
Weather
There is a 20% chance of excessive snow which would cause a delay of approximately 2
weeks which would cost £80,000.
Construction Materials
There is a 10% probability of the price of construction material dropping, which will save
the project £90,000.
Workers going on strike
There is a 5% probability of construction coming to a halt if the workers go on strike. The
impact would cause additional costs of £150,000.
Planning permission
There is a chance of 5% that the planning permission required will be returned 3 weeks
earlier than expected, which would create a saving of £115,000.
Which of these outcomes are risks and which are rewards?
Which has the highest expected monetary value?
Which has the lowest expected monetary value?
If all of the outcomes listed occur, would they represent a total loss or a total gain?
Activity adapted from http://www.brighthubpm.com/risk-management/48245-calculating-expected-monetary-value-emv/
The Further Mathematics Support Programme
Making decisions using mathematics
Expected Monetary Value (EMV) - Solutions
Weather
There is a 20% chance of excessive snow which would cause a delay of approximately 2
weeks which would cost £80,000. EMV = 0.2 x -£80,000 = -£16,000
Construction Materials
There is a 10% probability of the price of construction material dropping, which will save
the project £90,000. EMV = 0.1 x £90,000 = £9,000
Workers going on strike
There is a 5% probability of construction coming to a halt if the workers go on strike. The
impact would cause additional costs of £150,000. EMV = 0.05 x -£150,000 =- £7,500
Planning permission
There is a chance of 15% that the planning permission required will be returned 3 weeks
earlier than expected, which would create a saving of £115,000.
EMV = 0.15 x £115,000 = £17,250
Which of these outcomes are risks and which are rewards? Weather and workers striking are risks;
the other two bring possible rewards.
Which has the highest expected monetary value? The planning permission at £17,250
Which has the lowest expected monetary value? The weather at -£16,000 (i.e. a £16,000 loss)
If all of the outcomes listed occur, would they represent a total loss or a total gain?
£17,250 + £9,000 - £16,000 - £7,500 = £2,750 so a total gain (as the value is positive).
Activity adapted from http://www.brighthubpm.com/risk-management/48245-calculating-expected-monetary-value-emv/
The Further Mathematics Support Programme
Making decisions using mathematics
An Introduction to Decision Trees
A decision tree is used to show possible risks and rewards in a given scenario and
thereby calculate the best overall course of action. It looks a bit like a tree diagram, but
also incorporates the following symbols which are placed at the end of branches:
Decision node
Chance node
End node
Work in pairs to construct a decision tree for the following Dice Game.
See if you can work out the EMV for each stage of the game, and hence work out the
overall EMV.
What would you recommend? Would you play the game? If you get a 5 or a 6, should
you roll again?
In a game you are asked by the Gamekeeper to roll a fair dice.
If a 5 or a 6 is obtained, the Gamekeeper will pay you £20.
For any other number you have to pay the Gamekeeper £5.
However, in the second case, instead of paying £5 straight away you can opt to roll the
dice again. If you roll again and score a 6, the Gamekeeper will pay you £35.
Otherwise you lose a further £5 and so you will need pay £10 in total.
An Introduction to Decision Trees - Solution
The Dice Game
In a game you are asked by the Gamekeeper to roll a fair dice.
If a 5 or a 6 is obtained, the Gamekeeper will pay you £20.
For any other number you have to pay the Gamekeeper £5.
However, in the second case, instead of paying £5 straight away you can opt to roll the
dice again. If you roll again and score a 6, the Gamekeeper will pay you £35.
Otherwise you lose a further £5 and so you will need pay £10 in total.
Roll a 6
1
5
( × £35) + ( × −£10)
6
6
Probability
£35
1
6
-£2.50
Roll again
- £10
Roll 1 - 5
-£2.50
Probability
5
6
1 – 4 scored
2
Probability 3
Don’t roll
again
-£5
£5
Play the
game
5 or 6 scored
£20
1
Probability 3
Don’t
play the
game
£0
Work backwards from right to left, calculating the EMV for each decision or chance node. Place a
double line across rejected decisions.
The best strategy for this game is: Play the game, with overall EMV of £5.
If a 1-4 is scored, the best strategy is to roll again as this has a better EMV than not rolling again.