24.4.2

Let T1 be...

programming language R

BUFFON'S NEEDLE AND CROSS 545 24.4 Buffon's needle and cross The following question was first considered by George Louis Leclerc, later Comte de Buffon, in 1733: If a thin, straight needle of length l is thrown at random onto the middle of a horizontal table ruled with parullel lines a distance dl apart, so that the needle lies entirely on the table, what is the probability that no line will be crossed by the needle?" The answer depends on 1 and so simulation of this experiment offers a way of estimating 1. We will look at the complementary probability that the needle actually intersects with a ruled line on the table; call this a crossing. 24.4.1 Theoretical analysis We can think of the position of the needle as being determined by two random variables: Y : the perpendicular distance of the centre of the needle from the nearest line on the table and X : the angle that the top half of the needle makes with a ray through its centre, parallel to the table lines and extending in a positive direction. See Figure 24.9 for a sketch. For the position of the needle to be random, we require Y to be U(0,d/2) and X to be U(0,). We then define the sample space of all possible outcomes or positions of the needle as =[0,][0,d/2]. 1. Identify the inequality that X and Y must satisfy if the needle is to cross a ruled table line. Draw a picture of the sample space and use your inequality to shade that part of it that corresponds to a crossing. We will refer to this region as the crossing region C. 2. As the needle is thrown at random, the probability of falling in any region R in can be calculated as the ratio of the area of R, denoted R, to the total area of . That is, 2R/(d). Using integration, find the area of C and hence confirm that the probability of a crossing is 2l/(d). 24.4.2 Simulation estimates Let T1 be the number of crossings in n tosses of the needle, then E1=T1d/(nl) is an umbiased estimator of 2/. Calculate the variance of E1 and thus suggest the best needle length l to use, subject to the restriction ld. Using the value of l obtained above, write a program to simulate E1 using n=100,000 needle tosses