This problem solving activity has a statistics focus.
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 circle red; the first ring blue; and the outside ring 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.
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 circle red; the first ring blue; and the outside ring 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?
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.
The problem says that the darts land at random on the board. So the proportion of the darts that land in the yellow region should be the same as the proportion of the total target that is yellow.
The area of the whole target (a circle with a radius of 3m) can be found using the formula for the area of a circle (πr2). π x 3m2 = 28.27m2
The area of the part of the target that is not yellow is a circle with a radius of 2m. π x 2m2 = 12.57m2
Therefore 12.57/28.27, or 44% of the board is not yellow. Therefore we would expect 44 of the 100 darts to land on that part of the board, and the rest, 56 of the darts, to land in the yellow region.
Printed from https://nzmaths.co.nz/resource/another-dartboard at 12:46am on the 3rd May 2024