### Very Old Man

Is the age of this very old man statistically believable?

### Into the Exponential Distribution

Get into the exponential distribution through an exploration of its pdf.

### Into the Normal Distribution

Investigate the normal distribution

# Discussing Risk and Reward

##### Stage: 5 Challenge Level:

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?