Another Dartboard
Predict probabilities of an event involving area
Make deductions from probabilities
Devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, look for patterns, draw a diagram, make a table, think, use algebra).
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. So here we have to find out the relative areas of the board to determine the probabilities of the dart landing in a particular area.
This type of approach can actually be used for simulation. So we can do an experiment that will tell us something about a practical situation. Take areas under curves for instance. Suppose that we wanted to find the area between the curve y = x2 and the x-axis for x = 0 to x = 1. (See diagram.) Then 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 ≤ x2, 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 point is then, that the idea behind the problem that we have posed here can be used in a number of applications. It turns out that simulation is a very useful tool in mathematics/statistics and can be used in a large number of situations.
Problem
Lenette is building a dartboard. She decides to make one with three concentric circles. The smaller one has radius 1 metre; the next one 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.
Now Lenette is not a good darts player. She has no great control over the darts and they 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 Lenette 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 Lenette’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? - Get the students to tackle the problem in their groups.
- As the students work 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? - Give students time to write up their conclusions
- Allow two to three different groups to report back.
Extension
Construct Lenette’s board and test the probabilities you have predicted above.
You might do this by computer and so run a longer simulation than you could by hand.
| Attachment | Size |
|---|---|
| AnotherDartboard.pdf | 53.83 KB |
| AnotherDartboardMaori.pdf | 50.46 KB |
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