Buffon's Needle problem (Geo Prob)

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Discussion Overview

The discussion revolves around Buffon's Needle problem, specifically focusing on the interpretation and solution of question "A" or q.1. Participants explore the theoretical aspects of the problem, its mathematical formulation, and practical applications, including a request for an Excel model for calculations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant seeks clarification on question "A" of Buffon's Needle problem, indicating a lack of understanding.
  • Another participant points out that there is no question "A" and suggests it may refer to question 1 (q.1).
  • A participant provides a detailed explanation of how to approach q.1, suggesting that it involves assuming 1,000 needle drops with angles distributed evenly between 0 and π, and discusses the process of renormalization and area approximation.
  • Concerns are raised about the implications of using π in the approximation, questioning the assumptions required for the calculations.
  • A different participant mentions working on a practical application of Buffon's Needle for cases where the needle length exceeds the distance between the lines and requests an Excel model for probability calculations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of question "A" or the specifics of the calculations involved. Multiple viewpoints and approaches are presented without resolution.

Contextual Notes

There are unresolved assumptions regarding the distribution of angles and the implications of using π in the calculations. The discussion includes varying interpretations of the problem and its requirements.

Who May Find This Useful

Individuals interested in probability theory, mathematical modeling, or practical applications of geometric probability may find this discussion relevant.

nomi
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http://www.mste.uiuc.edu/reese/buffon/buffon.html#questions

can someone explain problem "A" for me please. this isn't for school or homework but just something that i don't understand and would like to know where I'm going wrong.

how would i go about solving question "A"?

thanks
 
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There is no question A. Did you mean q.1?
 
EnumaElish said:
There is no question A. Did you mean q.1?
yes that's what i mean
 
It seems to me that q.1 is asking you to assume 1,000 needle drops with thetas distributed evenly between 0 and pi.

First, renormalize by multiplying all areas by 2/pi. After renormalization the area of the large rectangle is 1 and the shaded area is 2/pi. Prob{hit} = (2/pi)/1 = 2/pi as before.

Now you are to approximate the shaded area by calculating the area for 1,000 little rectangles. The 1st little rectangle has area 0. The 2nd has area = base x height x renormalization = (pi/1000) x sin(pi/1000)/2 x 2/pi = sin(pi/1000)/1000. The 3rd has area = (pi/1000) x sin(2pi/1000)/2 x 2/pi = sin(2pi/1000)/1000. The Nth has area = (pi/1000) x sin((N-1)pi/1000)/2 x 2/pi = sin((N-1)pi/1000)/1000.

Sum area = [tex]\right.\sum_{N=1}^{1000}\sin\left(\frac{(N-1)\pi}{1000}\right)\left/1000[/tex]

It seems to me like this sum area is the approximate 2/pi that the question is after. (This still doesn't make a whole lot of sense to me because pi is parametric in the formula itself; so one must already assume an exact value for it before one can approximate it.)
 
I am working on a practical application of Buffon's Needle for when the length of the needle is greater than the distance between the lines.

I am looking for an EXCEL model to calculate the probability.

Thanking You in Advance.
 

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