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Chapter 6: Q.6.4 (page 275)

Solve Buffon鈥檚 needle problem when L > D. answer: 2L 蟺顿(1 鈭� sin 胃) + 2胃/蟺, where cos 胃 = D/L.

Short Answer

Expert verified

The Buffon's needle problem is proved where L > D.

Step by step solution

01

Content Introduction

Given a floor with evenly spaced parallel lines a distance d apart, Buffon's needle problem asks for the probability that a needle of length l would land on a line.

02

Content Explanation

When L > D, needle will surely cut at least one line for all 胃such that Lcos胃鈮�D

Therefore from 0to胃(such that cos胃1=DLneedle will surely cut one line at least. Also, will 胃be on both sides.

Therefore, out of 蟺angle available 2胃will give a sure event for angles more than胃.

03

Explanation Proof

P{X<L2cos胃=鈭�fX(x)f0(y)dxdy=4蟺顿鈭�胃1蟺2鈭�0L2cosydxdy=4蟺顿鈭�胃蟺2L2cosydy2L蟺顿(1-sin胃1

Total probability 2L蟺顿(1-sin胃1+2胃蟺

where cos胃1=DL

Hence proved.

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