Making decisions using mathematics Introduce the idea of decision making using mathematics rather than just human judgement. 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…. PPT image taken from http://www.projectmanage.com/files/making_decisions.png

Making decisions using mathematics 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

Making decisions using mathematics 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? This slide is just for interest and can be deleted if preferred. The number of combinations is 49C6 = 13, 983, 816. Encourage students to use the nCr button their calculator.

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) Give students the EMV handout.

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. 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: 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?

Decision Trees decision node chance node end node Give out decision trees handout. Discuss the meaning of each of the node types and how they are used on a decision tree.

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. 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. 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. After reviewing the Dice Game, pairs of students could either pick a task from the Decision Trees Groups Tasks handout or tasks could be allocated by ability (Tasks B and D are probably the most demanding). Bring the group back together to discuss the tasks after they are completed, 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.

Dice Game Solution If decide to play, you might roll a 5 or 6 , or roll 1-4 For the calculations, work backwards 5 or 6 (p = 1 3 ) 1 – 4 (p = 2 3 ) £20 First decision is play or don’t play A double line goes on the branch for the least favourable option Roll again Don’t roll again 6 (p = 1 6 ) 1-5 (p = 5 6 ) £35 -£10 -£5 - £2.50 Play Don’t play £0 £5 £5 - £2.50 Calculations are (right to left): 35× 1 6 + −10× 5 6 =-£2.50 which is preferable to -£5 on the other branch of this pair, so choose -£2.50 1 3 ×20 + 2 3 ×−£2.50 =£5 and this is preferable to not playing, which has EMV of £0. Hence play the game and if you get 1-4 on the first roll, choose to roll again. Overall EMV of the game is £5. 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

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! To close the session give students an overview of the role of risk analysis in business and as a career.

And remember….. http://www.enterpriseirregulars.com/wordpress/wp-content/uploads/2013/10/keep-calm-and-make-educated-decisions-2.png

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. The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students. To find out more please visit www.furthermaths.org.uk