Dropping needles to estimate pi
A floor is marked with parallel lines a distance d apart. A needle of length d (the same as the spacing) is dropped so that its position and orientation are uniformly random.
- What is the probability that the needle crosses one of the lines?
- Explain how dropping many needles gives an estimate of pi, and why the estimate converges slowly.
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Describe the needle by the distance from its centre to the nearest line (between 0 and d/2) and the angle it makes with the lines (between 0 and a right angle). It crosses a line when the needle's projection perpendicular to the lines reaches the nearest line.
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