Answer Happy

Question 6 [10 Marks] Buffon's needle is a famous problem in probability. Consider a needle dropped onto floorboards. The problem is to determine the probability the needle will lie across a line between two floorboards. Suppose the needle has length 1 and the floorboards are of width d> 1. For simplicity assume that each board runs the full length of the floor, so all lines between floorboards are parallel. a) Perform an experiment to estimate the probability a needle dropped onto such a floor crosses a line between floorboards. (Note you do not need actual floorboards – you can rule parallel lines on a sheet of paper.) Describe your experimental procedure and results in full detail. b) It can be shown that the exact probability of crossing a line is 21/(dr). Use this fact and your results from part a) to estimate n. c) Does your estimate of a improve if you drop more needles? To investigate this, repeat parts a) and b) with different numbers of needles (or one needle dropped different numbers of times). Describe the relationship between accuracy and sample size for this experiment, using a graph to illustrate your answer. d) [Optional for bonus marks – 2 marks] What sample size would be needed to estimate a correct to 2 decimal places? (We will learn precise methods for calculating sample sizes later in the unit – here you should base your answer on the relationship you described in part c.)