Solution 1: Buffon's Needle Problem

Solution 1: Buffon's Needle Problem.

The solution is quite similar to that given for the special case when L < d. Now suppose L d. For example, if d = 3 and L = 5, then the sample space is the square

(shaded grey and red below) and the event space is the region that lies between the curves

(shaded red below).

The probability is the area of the red region divided by the area of the grey and red regions. Problem 1 asks for a general solution, which should expressed in terms of L and d. The probability we seek is equal to:

The best way to calculate the area of the red region is to integrate with respect to y. First let's divide the red region into two equal parts by drawing the line = / 2. Then we may multiply the red area left of this line by two, in order to get the entire area between the two curves.

Because we are integrating with respect to y, we must solve our equations in terms of . Recall from calculus that to find the area of this region, we take the integral of the right curve, = / 2, minus the integral of the left curve, . Note that y varies from 0 to d, which are the limits of the integral. Using this formula, our probability can be calculated from the following equation:

EXERCISES:

Solve Buffon's coin problem with rectangular tiles that have height h and width w.
Solve Buffon's coin problem with equilateral triangles that have side length 1.
Problem 10 Supplementary Exercises pg 29: NCTM
Find the probabilities of the following events in Buffon's needle experiment. Let d = 1. In each case, sketch the event as a subset of the sample space.
1.) Suppose L < d. {0 < X < / 2, 0 < Y < 1 / 3}
2.) Suppose L < d. {1/4 < Y < 2 / 3}
3.) {X < Y} Hint: Solutions

Problem 1

References: Virtual Laboratories in Probability and Statistics; NCTM