Buffon's Needle problem (Geo Prob)

[nomi]

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

 

[EnumaElish]

There is no question A. Did you mean q.1?

 

[nomi]

There is no question A. Did you mean q.1?

yes that's what i mean

 

[EnumaElish]

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 = $$\right.\sum_{N=1}^{1000}\sin\left(\frac{(N-1)\pi}{1000}\right)\left/1000$$

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.)

 

[TheRumpledOne]

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.