Ngaio makes a dartboard with three concentric circles.

The smallest one has radius 1 metre; the next has radius 2 metres; and the third has radius 3 metres.

She paints the centre circular area red; the annulus between the small circle and the next circle she paints blue; and the rest of the board she paints yellow.

Ngaio is not a good darts player and her darts land at random on the board.

If she throws 100 darts, about how many would you expect to land in the yellow region?

What scoring system would you use for Ngaio if you wanted her, on average, to get the same score each time she threw 3 darts?

This problem is about an appreciation of probability. A dice has six sides and every one is equally likely to land facing up. Hence the chances of any one particular face showing is one out of six or 1/6. For a dart randomly landing in an area, the probability of it landing in a given area is in proportion to the size of the area. In this problem students must find the relative areas of the board to determine the probabilities of the dart landing in a particular area.

This approach can be used for simulation. An experiment can be done that will tell us something about a practical situation.

For example: Suppose that we want to find the area between the curve y = x^{2} and the x-axis for x = 0 to x = 1. (See diagram.) One way to do this is to use simulation.

Imagine throwing a dart at random at the square. It will land on a point with co-ordinate (x, y). If y ≤ x^{2}, then the point is in the area we want. If not, it is in the top half of the square. If we then count the proportion of times that the dart hits the area under the curve, we know the proportion of the square that is occupied by the area under the curve. Hence we can find an estimate for the area under the curve. Obviously the more ‘darts’ we throw, the more accurate this estimate will be.

The ideas embedded in this problem can be used in a number of applications. Simulation is a useful tool in mathematics/statistics and can be used in a large number of situations.

### Problem

Ngaio makes a dartboard with three concentric circles. The smallest one has radius 1 metre; the next has radius 2 metres; and the third has radius 3 metres.

She paints the centre circular area red; the annulus between the small circle and the next circle she paints blue; and the rest of the board she paints yellow.

Ngaio** is not** a good darts player and her darts land at random on the board. If she throws 100 darts, about how many would you expect to land in the yellow region?

What scoring system would you use for Ngaio if you wanted her, on average, to get the same score each time she threw 3 darts?

### Teaching sequence

- Revise the idea of probabilities by talking about dice and coins and the probability of an event using them.
*What is the probability of getting a 5 when you throw a dice? Why?*

What is the probability of getting two heads in two throws of a dice? Why? - Pose Ngaio’s problem.
- Ask the class how they would find probabilities where are concerned.
*What are the important things that you need to know?*

How would you go about finding these things? - As the students work in groups, ask questions that focus on their understanding of probability.
*Is any one spot on the board more likely than another?*

Which region are you most likely to land in? Why?

How did you work that out?

Are you convinced that you are correct? Why/why not? - Allow students time to write up their conclusions
- Have groups share their solutions.

#### Extension

Construct Ngaio’s board and test the probabilities you have predicted above.

You might do this by computer and so run a longer simulation than you can by hand.