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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 ...
Discussing Risk and Reward
Age 16 to 18
Challenge Level 
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 ...
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