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Here is a way to solve Buffon's needle problem without calculus. Recall that an inch needle is dropped at random onto a lined sheet, where the lines are one inch apart. (a) Let A be the number of lines that the needle hits. Let B be the number of times that a polygon of perimeter hits a line. Show that E[A] = E[B]. (Hint: Use linearity of expectation.) (b) Assume that l< 7. Calculate the expected number of times that a circle of perimeter l hits a line. (c) Assume that < 1. Use part (a) and (b) to derive a formula for the probability that the needle hits a line. (Hint: The number of hits is a Bernoulli random variable.)

Here is a way to solve Buffon's needle problem without calculus. Recall that an inch needle is dropped at random onto a lined sheet, where the lines are one inch apart. (a) Let A be the number of lines that the needle hits. Let B be the number of times that a polygon of perimeter hits a line. Show that E[A] = E[B]. (Hint: Use linearity of expectation.) (b) Assume that l< 7. Calculate the expected number of times that a circle of perimeter l hits a line. (c) Assume that < 1. Use part (a) and (b) to derive a formula for the probability that the needle hits a line. (Hint: The number of hits is a Bernoulli random variable.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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5. Here is a way to solve Buffon's needle problem without calculus. Recall that an l inch needle is dropped at random onto a lined sheet, where the lines are one inch apart. (a) Let A be the number of lines that the needle hits. Let B be the number of times that a polygon of perimeter l hits a line. Show that E[A] = E[B]. (Hint: Use linearity of expectation.) (b) Assume that l < T. Calculate the expected number of times that a circle of perimeter l hits a line. (c) Assume that l < 1. Use part (a) and (b) to derive a formula for the probability that the needle hits a line. (Hint: The number of hits is a Bernoulli random variable.)
Transcribed Image Text:5. Here is a way to solve Buffon's needle problem without calculus. Recall that an l inch needle is dropped at random onto a lined sheet, where the lines are one inch apart. (a) Let A be the number of lines that the needle hits. Let B be the number of times that a polygon of perimeter l hits a line. Show that E[A] = E[B]. (Hint: Use linearity of expectation.) (b) Assume that l < T. Calculate the expected number of times that a circle of perimeter l hits a line. (c) Assume that l < 1. Use part (a) and (b) to derive a formula for the probability that the needle hits a line. (Hint: The number of hits is a Bernoulli random variable.)
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