GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team 2 Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team 6 Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
Risk and Reward (Teacher s notes) These notes are intended to assist the teachers working with groups of students at a table, they give an overview of the lesson and suggestions on how to get the most from the activity. Each table will need: A six sided die One set of student sheets per student Short introduction : One of the aims of the new Core Maths qualifications is to show how mathematics is used almost every day, in work and other aspects of our lives. This activity shows just one situation, there are many more which use exactly the same mathematical ideas. The theme of this activity is risk and reward. It is a game designed to make you think about how you make decisions which involve an element of risk. Mathematicians often use games to investigate how people react in situations and decide on winning strategies. Although it can mean other things, a winning strategy generally means the one which we gain the most from, for example in business this could be the most profitable. The activity will provide opportunities to discuss common misunderstandings such as Gambler s Fallacy, and think about the model and the assumptions, for example: Is it fair to assume that the will always catch a lobster or be washed away in bad weather? Will the probability of good weather increase in summer and decrease in winter? Introducing the game Although it is possible for everyone to play as individuals, playing as pairs makes the task of playing and recording easier and tends to increase discussion when deciding on where to place the. Run through the instructions getting the students to fill in the sheet ready for the first day. (See the students sheet(s) at the end of this document). When everyone has made the decision of where to place their for the first day, roll the die and determine the weather, it is probably better if you do this. Play the game for a short time, aim to allow the students time to experience the process of decision making. (-0 rounds) Some things to look out for whilst playing the game: Are they quite risk averse at the start but gradually take more risks as the game progresses? How do students react after a good round/bad round? Sometimes, after a bad round, students take more risks to try and gain a larger profit on the next round. Sometimes students also take more risks after a good round because they feel they are playing with winnings rather than their own cash. You could discuss the pitfalls of such approaches. Comments like It s been good weather for the last three or four rounds, it s more likely to be bad weather on the next round. Although over a long run the weather should balance out to the expected probabilities this does not
Number of days influence individual rounds. Just like when flipping a coin, we expect roughly equal amounts of heads and tails, but on each flip the probability of heads remains ½. Discussing the game Aim to bring out the points mentioned above under some things to look out for. At some point before introducing the weather simulation data the person leading the session will attempt to bring things together. It may be possible for the presenter to quickly collect data from each group and arrive at a figure for each of the number of good days, bad days and hurricanes, in total. The presenter will then introduce the idea of using a computer model to gather data for a year. The presenter will then suggest working with the simulated weather data and work will resume at tables. Then introduce the weather simulation data. Use the first question and the prompt on the student sheet to encourage the students to consider how this data could be used by the lobster pot businesses. Give them time to think and offer suggestions. You are looking to prompt them towards working out the anticipated profits over the course of a year for an offshore and inshore pot. For example using the data in the first diagram on the weather simulation data sheet see below. Simulated For Year 0 00 20 200 0 00 0 0 290 69 Bad Days Good Days Hurricanes There are 69 bad days, 290 good days, and hurricanes, on the 69 bad days an offshore pot will be lost making a loss of 6, or profit of - 6, over the simulated year this would amount to 69 x - 6 which is -. On the 290 good days an offshore pot makes per day making a total of 290 x which is 0. Finally on the days when there is a hurricane a loss of 6, or profit of - 6, is incurred on each of the days making a total loss of 0. If a pot was placed offshore every day for a year then the business would anticipate earning (69 x - 6) + (290 x ) + ( x - 6) = 006
The inshore pot makes on a good day, on a bad day and - 6 when there is a hurricane, so for an inshore pot, the profit for year would be (69 x ) + (290 x ) + (x- 6) = 88 In this year placing the pot inshore would have been more profitable. Here are the values for all simulations Year Year 2 Year Year Bad Days 69 70 7 6 Good Days 290 29 0 00 Hurricane Offshore Profit,006,09,60,0 Inshore Profit,8,22,79,8 Finish by considering the final question. Would this strategy win every single round? You are aiming to get them to see that although the earnings over a year tends to be better for the inshore, each individual round is subject to random chance. It is easy to check back and see how many times the offshore were better in the previous gameplay. Ask the students to look back at their game and see. The presenter will at an appropriate point gather the class together to round off the session. If your students have reached this stage please consider the extension below. Extending the task On the Year simulation (7 bad days, 0 good days, and hurricanes) the anticipated earnings from inshore and offshore were almost equal. Challenge the students to find a set of data that make the anticipated earnings from inshore and offshore equal. Is this possible? Is it likely? Are there more than one set of values? Thinking solution As hurricanes affect the income from inshore and offshore in same way, the number of hurricanes can be ignored. For every good day the offshore pot gains an extra 2 over the inshore pot. (ie offshore, inshore ) For every bad day the offshore loses to the inshore pot. (ie offshore - 6, inshore ) This means for every bad day the offshore pot needs. good days to stay on equal terms. Providing that this condition is met the values of both should be the same. The number of hurricanes would be 6 (good days + bad days). For example, 6 good days 08 bad days and one hurricane. Within the constraints of the problem this means the bad days must be an even number, and the combined total of good days and bad days cannot exceed 6, one year.
Using algebra a = good days b = bad days and c = hurricanes we need a -6b 6c = a + b 6c a -6b = a + b 2a = b a =.b
Student sheet Lobster Pots Lobster - is a game which can be used to think about risk and reward. Your business involves fishing for lobsters; these can be sold for a good profit at the local fish market. Sometimes fishing in more risky areas can bring greater rewards but there are also potential problems. Each player starts with a boat and 6 lobster. The can be placed either inshore or offshore, and left overnight to catch lobsters. If the weather is good, you can expect to catch one lobster per pot, both inshore and offshore. As the offshore lobsters are bigger and of better quality, they fetch more at the fish market, each compared with for the inshore lobsters. However, if the weather is bad the offshore get washed away, they cost 6 each to replace. In bad weather, only the inshore catch lobsters. As fewer lobsters are available, the price of an inshore lobster increases from to. The weather is normally good ( out of 6 days). If the weather is bad on one day then the probability of getting good weather on the following day falls to out of 6. Three bad days in a row result in a hurricane and all of the, inshore and offshore, are destroyed. Playing the game. Each Team starts with 0 and 6 lobster. 2. For the first round assume the weather on the previous day was good.. Decide how many lobster you wish to place inshore and how many you wish place offshore.. Fill in columns and 2 in the game table below.. Roll the dice to decide the overnight weather conditions. If the weather on the previous day was good, rolling (,2,,, or ) gives good weather, a 6 gives bad weather. If the previous day was bad weather, rolling (,2,, or ) gives good weather, a ( or 6) gives bad weather. Three consecutive days of bad weather produces a hurricane on the third day and all the, both inshore and offshore, are lost and must be replaced. 6. Calculate how much each team has earned/lost and complete the row. 7. Repeat instructions to 6 until finish time is called.
GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team 2 Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
GAME TABLE :Team Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0 GAME TABLE :Team 6 Day Inshore Offshore Overnight Income (I) Costs (C) Profit (I-C) Running total 2 6 7 8 9 0
Number of days Number of days Number of days Number of days simulation data This weather data was produced by a spreadsheet which simulated playing for 6 days. Simulated For Year Simulated For Year 2 0 0 00 290 00 29 20 20 200 200 0 0 00 69 00 70 0 0 Bad Days Good Days Hurricanes 0 0 Bad Days Good Days Hurricanes Simulated For Year Simulated For Year 0 00 0 0 00 00 20 20 200 200 0 0 00 0 0 7 Bad Days Good Days Hurricanes 00 0 0 6 Bad Days Good Days Hurricanes How could this data be used by the lobster pot players? Use the data to suggest a strategy which could prove successful if the game were played for 6 rounds. Would this strategy win every single round?
keith.proffitt@mei.org.uk
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Statistical and financial modelling
Modelling cycle problem???? problem???? good enough? solution solution PPPP
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The rules
Your data Table Good days Bad days Hurricanes 2 6 7 8 9 0 Totals 0 0 0 How could this data help you to decide where to place your? Would the data be exactly the same if we played the game again?
Text simulation for a year
Exam question
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6.0.0 CPI inflation (%).0.0 2.0.0 0.0 Jun- Sep- Dec- Apr-2 Jul-2 Oct-2 gallon =. litres Mr Green drove 67 miles on business last year. His car does an average of 2. miles per gallon. Unleaded petrol, per litre (p) Estimate how much Mr Green will spend on petrol for business journeys next year. 0 0 Jun- Sep- Dec- Apr-2 Jul-2 Oct-2
Financial modelling - mortgage
Student sheet Lobster Pots Lobster - is a game which can be used to think about risk and reward. Your business involves fishing for lobsters; these can be sold for a good profit at the local fish market. Sometimes fishing in more risky areas can bring greater rewards but there are also potential problems. Each player starts with a boat and 6 lobster. The can be placed either inshore or offshore, and left overnight to catch lobsters. If the weather is good, you can expect to catch one lobster per pot, both inshore and offshore. As the offshore lobsters are bigger and of better quality, they fetch more at the fish market, each compared with for the inshore lobsters. However, if the weather is bad the offshore get washed away, they cost 6 each to replace. In bad weather, only the inshore catch lobsters. As fewer lobsters are available, the price of an inshore lobster increases from to. The weather is normally good ( out of 6 days). If the weather is bad on one day then the probability of getting good weather on the following day falls to out of 6. Three bad days in a row result in a hurricane and all of the, inshore and offshore, are destroyed. Playing the game. Each Team starts with 0 and 6 lobster. 2. For the first round assume the weather on the previous day was good.. Decide how many lobster you wish to place inshore and how many you wish place offshore.. Fill in columns and 2 in the game table below.. Roll the dice to decide the overnight weather conditions. If the weather on the previous day was good, rolling (,2,,, or ) gives good weather, a 6 gives bad weather. If the previous day was bad weather, rolling (,2,, or ) gives good weather, a ( or 6) gives bad weather. Three consecutive days of bad weather produces a hurricane on the third day and all the, both inshore and offshore, are lost and must be replaced. 6. Calculate how much each team has earned/lost and complete the row. 7. Repeat instructions to 6 until finish time is called.
Number of days Number of days Number of days Number of days simulation data This weather data was produced by a spreadsheet which simulated playing for 6 days. Simulated For Year Simulated For Year 2 0 0 00 290 00 29 20 20 200 200 0 0 00 69 00 70 0 0 Bad Days Good Days Hurricanes 0 0 Bad Days Good Days Hurricanes Simulated For Year Simulated For Year 0 00 0 0 00 00 20 20 200 200 0 0 00 0 0 7 Bad Days Good Days Hurricanes 00 0 0 6 Bad Days Good Days Hurricanes How could this data be used by the lobster pot players? Use the data to suggest a strategy which could prove successful if the game were played for 6 rounds. Would this strategy win every single round?