2. Making decisions using mathematics

4. What are the chances of choosing the winning set of numbers for the National Lottery? How many possible combinations of 6 numbers can be chosen from 49 numbers?

5. 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)

6. 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.

7. Decision Trees decision node chance node end node

8. How to use a decision tree Kevin has £5000 which he might invest in a new business scheme. There is a probability 0.3 of a 5% profit, a probability 0.2 of a 7% profit and a 0.5 probability of a 2.5% loss. Should Kevin invest his money in this scheme?

9. How to use a decision tree - solution £5250 (profit 5%) £5350 (profit 7%) £4875 (loss 2.5%) Box indicates a decision to be made Circle indicates chance Triangle indicates a final outcome £0 Invest Don’t invest 0.3 0.2 0.5 All possible events that can happen by chance (0.3x£5250)+(0.2x£5350)+(0.5x£4875) = £5082.50 £5082.50 Double line to cross out least favourable option EMV is £5082.50 so Kevin should invest!

10. 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 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.

11. Roll again Don’t roll again £35 -£10 -£5£20 Play Don’t play £0 Dice Game Solution £5 - £2.50 £5 For the calculations, work backwards First decision is play or don’t play If decide to play, you might roll a 5 or 6, or roll 1-4 If you roll 1-4, the decision is then whether to roll again, in which case you might score a 6 or score 1-5 A double line goes on the branch for the least favourable option

12. Risk Analysis…a career  Well suited to graduates of mathematics, business, accountancy or statistics.  Starting salary of over £21,000  Salary of between £29,000 and £44,000 after six years  Risk managers can earn in excess of £70,000  Opportunities in a range of fields including banking, large credit organisations, analysing risk of changes in the law, insurance and many others!

13. And remember…..

14. The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit www.furthermaths.org.uk www.furthermaths.org.uk The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.