Buffon Needle Problem

Transcription

1 Buffon Needle Problem MATH 171 Freshman Seminar for Mathematics Majors J. Robert Buchanan Department of Mathematics 2010

2 History Georges Louis Leclerc (Comte de Buffon, ) was a French naturalist (general scientist, mainly biologist). In 1777 he devised a probabilistic means of approximating π.

3 The Problem Toss a needle which is l units long and you toss it at random onto a plane containing parallel lines spaced a > l units apart.

4 Simulated Result

5 Question What is the probability that a needle crosses one of the horizontal lines?

6 Question What is the probability that a needle crosses one of the horizontal lines? Perform 100 tosses and record the number of times the needle crosses the line. The toothpicks have a mean length of mm. The spacing between the lines is mm.

7 Theoretical Result Now that we have some data we should compare it against theory. Let the distance from the midpoint of the needle to the nearest horizontal line be y. Let the angle the needle make with the horizontal line be θ.

8 Theoretical Result Now that we have some data we should compare it against theory. Let the distance from the midpoint of the needle to the nearest horizontal line be y. Let the angle the needle make with the horizontal line be θ. 1 Under what conditions on l, y, and θ will a needle cross a horizontal line? 2 What are the minimum and maximum distances of the midpoint of the needle from the nearest line? 3 What are the minimum and maximum angles that the needle can make with the lines?

9 Picture nearest line y Θ needle

10 Results 1 Under what conditions on l, y, and θ will a needle cross a horizontal line? y < l 2 sin θ

11 Results 1 Under what conditions on l, y, and θ will a needle cross a horizontal line? y < l 2 sin θ 2 What are the minimum and maximum distances of the midpoint of the needle from the nearest line? 0 y a 2

12 Results 1 Under what conditions on l, y, and θ will a needle cross a horizontal line? y < l 2 sin θ 2 What are the minimum and maximum distances of the midpoint of the needle from the nearest line? 0 y a 2 3 What are the minimum and maximum angles that the needle can make with the lines? 0 θ π

13 Summary The crossing status of a needle is completely determined if we know the y coordinate of its midpoint and the angle θ the needle makes with the horizontal.

14 Summary The crossing status of a needle is completely determined if we know the y coordinate of its midpoint and the angle θ the needle makes with the horizontal. Graph y = l sin θ in the rectangular window where 0 θ π 2 and 0 y a 2.

15 Graph 35 y Express the probability of a needle crossing a horizontal line in terms of the areas of regions in the graph above. Θ

16 Probability The theoretical probability that a needle crosses a horizontal line is the ratio of the area of the area under the graph of y = l 2 sin θ to the rectangle with dimensions π (width) and a 2 (height). P = l π 2 0 sin θ dθ aπ 2 = l 2 (2) aπ = 2l 2 aπ

17 Homework From the pooled data collected in class estimate the value of π. From the pooled data collected in class estimate the probability that a randomly chosen chord of a circle is longer than the side of an inscribed equilateral triangle.