Discussing risk and reward
10 starting points for risk vs reward
Age
16 to 18
Challenge level
Problem
Risk and reward are fundamental concepts in probability and chance: the more risky something is, the more reward you demand for taking part in that activity.
Here are some starting points for discussion or thought about risk, probability and reward. As you consider each point, try to validate your points clearly using well reasoned arguments or equations. Note that there is no 'right' answer to some of these questions. You might wish to find someone mathematically minded and debate these questions with him or her.
1. A lottery ticket costs £1, and 45% of the winnings are paid out in prizes. So, the fair price of a lottery ticket should be 45p.
2. A friend invites you to play a game where you roll a die. If a 6 comes up you win a prize of £1. How much would you pay to play this game a large number of times?
3. The same friend from question 2 adjusts the game so that if you roll a 6 then you win £1, but if you roll a 1 you have to pay an additional £1. How much would you pay to play this game 100 times?
4. A raffle is being held. There are 1000 tickets and a single top prize of £1000. How much would you pay for a ticket?
5. A lottery is being held in which there is a top prize of £X and 10X tickets are to be sold. For what range of X would you be prepared to spend £1 on a ticket?
6. Two treatments for a presently untreatable disease are under development. Treatment A is ambitious: If the treatment is successfully developed, it would cure all patients with the disease. This treatment will take 10 years to develop but has a 50% chance of failing its development. Treatment B is less ambitious: If the treatment is successfully developed it would cure 50% of patients with the disease. This treatment will also take 10 years to develop, but the programme is 90% likely to end up with a successful treatment. Which would you back? What data would you need to make your decision?
7. You buy a house and are offered two mortgage products. The first will cost you £1000 per month for 20 years, and is guaranteed to pay off your mortgage at the end of the 20 years. The second will also cost you £1000 per month for 20 years, but this will pay off the mortgage and pay you a bonus sum of £50,000 at the end of the 20 years with a 95% probability. However, there is a 5% chance that the investment will go sour and you will lose everything at a random point during the 20 years. Which product would you pick?
8. A certain physical activity is said to be highly exhilarating, but comes equipped with a 1 in 10,000 chance of breaking your leg. Would you take part?
9. Can you think of situations taught in different school subjects in which a risk is balanced by a reward?
10. Consider the challenge of inventing a way to 'measure' the risk associated with various physical activities, such as motorbike riding, walking, smoking or running whilst holding scissors. Once you have a system, why not try plotting these on a graph?
Getting Started
Use a mixture of common sense and mathematics.
Don't forget that risk is a personal preference -- you cannot be 'wrong' if you choose to accept a certain level of risk in an activity (although you might be foolish; but that's a different matter!)
Student Solutions
There are not necessarily 'right' or 'wrong' answers to various parts of this problem. 'Fair' prices for games involving chance are those for which the 'average' win matches the 'average' fee. However, risk-preference, utility and price of failure enter into all of these calculations. However, once we specify our position regarding these variables we can use probability to determine sensible prices.
Special mention goes to Jonathan from Nanjing International School: he tried many parts of this problem, and gave all of the 'fair' prices for the games.
1. In this question the expected winnings are 45p in the game. It could therefore be argued that the lottery is not a fair game, as you expect to lose a portion of any stake you place. However, to many people the concept of a big win is useful and this, therefore, warrants the unfair payout.
2. This question notes that you are to play the game a large number of times, so we can use averaging to determine the fair price: we must win, on average, at least as much as we pay for this game to make any long-term sense. For this game to be fair note that we will win £1 every 6 throws, on average. So, we should pay £1 for each 6 throws, which is about 17p per throw.
3. The expected payout for this game is zero: you win £1 1/6th of the time and forfeit £1 1/6th of the time. Therefore, the fair price for the game is £0. Why might someone pay to play this game? There is a 50% chance that you will win some amount of money between £0 and £100, and this might be of interest to some players.
4.The fair price for this game is £1, as there is a single prize of £1000 for 1000 tickets. However, many questions are raised by this: will all tickets be sold; how 'useful' would £1000 be to you - would this warrant an increased entry fee on your part to give you the chance of a win? This is very similar to the lottery where the 'utility' of the prize allows poor odds to stand.
5. This question appears to be very similar to question 4. However, it does raise starkly the issue of the utility of a very large prize. Imagine, for example, that the top prize would be £100,000,000. Many people would be tempted to spend £1 just to be in with the slimmest chance of winning this otherwise entirely unattainable sum of money.
6.This question raises the issue of the 'price of failure': in certain circumstances, it is OK if something does not work out; in other circumstances failure must be avoided at absolutely all costs. To make an informed decision about this question we would need to determine, in some sense, the 'price of failure' for the disease in question. Also, we might ask: is a 50% cure rate significant? For very virulant, contagious diseases this might be insufficient to stem an outbreak. However, if the disease leads to death, as opposed to a period of illness from which victims recover, the 50% cure rate suddenly seems very significant and appealing.
7. This question raises the idea of 'risk preferences'. To most people, the prospect of losing their house is not acceptable, and a 5% chance would seem very large in this context. The only sound advice to most people would be to choose the first product. However, for people with large numbers of investments the second option might seem appealing: there is a 5% chance of losing all, but a 95% chance of gaining £50,000. In order to make a decision an investor would need to work out the expected profit along with maximum loss. The fact that the default could occur at any time makes the computation more complex, but working on a default point, on average, at 10 years we see that
$$\mathbb{E}(\mbox{profit}) = 0.95 \times 50,000 - 0.05 \times1000\times 12\times 10 = 47500-6000 = 41500$$
On average, the payout is £41500 for the second product. This is a good deal in the context of a wider portfolio of investments.
Jonathan said: The first one, because in the second one there is a 1% chance that it'll go sour at 15years+ and I'll lose £15000- £19000
8. Jonathan said he would take part, and Emily said:
Yes, I would participate. Even if it was a 1 in 1000 chance of breaking my leg, I would do it. This is because I believe that living life without taking risks is like eating if you don't have a stomach-there's no point. If you don't take the risks, you won't live life to the fullest extent. If you don't take the risks, there's no thrill, there's no rush, you always know what's going to happen. To me, that would be worse than being dead. I love taking chances, especially when it's exhilarating.
Some people take risks, others don't; there is no right or wrong here. However, health and safety laws would deem this risk far too high a risk of harm for any activity that the public might pay to take part in.
9. Jonathan pointed out that usually in schools the main idea is that fair games are of the form "Unless the risk is 50% and the reward is 1:1" don't take it.
10. We'll leave this part open for consideration ...
Teachers' Resources
Why do this problem?
Risk, reward and chance are fundamental concepts which really
start to come alive during A-level and beyond; this collection of
short questions and discussion points provides many
opportunities to stimulate conversation and interest amongst a
class and bring elements of statistics to life.
Statistics in school can sometimes seem full of computation
and resources such as this one can provide the insights and sense
of purpose which might be obscured by the complicated
procedural aspects of statistics.
It can be used at a wide range of levels and
sophistications.
Possible approach
This problem can be used at various places in the curriculum
and it can be used for a sequence of short lesson starters across a
term, studied in more focus during key lessons or used to liven up
a lull in a lesson.
However, before first use, it is a good idea to get the idea
of risk-reward across to the group and stress the point that there
is not necessarily a 'right' answer to some of the questions.
Here are a few usage
suggestions:
- Give out all parts of the problem and let students discuss those points which catch their interest. After a while share comments and thoughts about the mathematics and issues which arose.
- Choose specific parts of the question and ask certain students to argue in favour of it and others against it. Ask for volunteers to 'debate' it in front of the class. You could split the questions between different groups so the audience hears debate on questions that they have not considered themselves in detail. During the debate, really focus the minds of the students on the mathematical clarity of the arguments. Unclear, vague arguments should be picked up on.
- Hand out the questions and ask the students to identify what mathematics and statistics are relevant to the question
- Require the students to compute numerical answers to many of the questions.
- Put this problem up as a poster for students to reflect on throughout the term.
Key questions
Who here is a risk taker? Who here is more of a cautious
person? Why?
What would prompt you to do something risky?
What mathematics is associated with risk and
probability?
When discussing these concepts, are you convinced that your
explanations are clear and precise?
Possible extension
This activity can be considered at sophisticated levels and
there are many possible lines of investigation which might arise.
Encourage gifted students to pursue lines of enquiry which seem of
interest. Consider, in particular, point 10.
Possible support
Some students might not perceive this type of activity as real
maths. Reassure them that it is a necessary part of developing
statistical skill and intuition to consider carefully activities
such as this one.