### 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

### 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.

You can read about some of the issues which might arise when teaching probability in this article.