### Why do this problem?

This

problem
is based around understanding the probability density function for
the normal distribution. The aim is to draw the learner into an
understanding of the properties of pdfs without requiring too many
complicated calculations: it uses and will reinforce ideas about
functions, integration and areas and the use of tables to calculate
the probabilities for standardised normal distributions. It will
also suit self-motivated independent learners.

### Possible approach

This question could sensibly be used once students are
starting to learn about the use of normal distribution tables and
standardised normal distributions. There is a lot of scope for
numerical estimation of probabilities and the first part could be
used to reinforce the fact that a probability density function
tells us quite a lot about a distribution even without the need for
complicated calculation. It will tie in nicely with other parts of
the syllabus on numerical integration.

### Key questions

What do you know about the area under a pdf?

What does the area under a pdf between two points mean?

How might we write down our probabilistic statements in terms
of standardised normal variables?

### Possible extension

Can learners find any of the areas enclosed by the lines in
the diagrams (using normal distribution tables).

Can they find the points of intersection on the graph?

### Possible support

Encourage learners to rely on their intuitive underestanding
of integration in terms of area. Alternatively, focus on the last
two parts of the question as a discussion. If they can't come up
with their own suggestions of calculation, perhaps they might
initially check the estimates of others?