- A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.
If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won.
By considering the possible positions for the centre of the disc,
- show that the probability of winning a prize on any particular roll is \(\frac { 1 } { 5 }\)
A group of 15 students each roll the disc onto the grid twenty times and record the number of times, \(x\), that each student wins a prize. Their results are summarised as follows
$$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
- Find the standard deviation of the number of prizes won per student.
A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
- Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students.
The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
- Explain how this probability could be used to find an estimate for the value of \(\pi\) and state the value of your estimate.