Mathematical problem about Buffon's Needle

[frostysh]

Buffon's Needle

A floor is ruled with equally spaced parallel lines a distance
D apart. A needle of length L is dropped at random on the floor. It is assumed that L no more than D. What is the probability that the needle will intersect
one of the lines? This problem is known as Buffon’s needle problem.

In summary, I think I spent on this problem a few month, trying to solve it, but suddenly without much of success. For now I have no sleep at night, and I have found some solution, unfortunately this solution is seems to be wrong, I have checked it with a book solution, and me is very sad and crying about that :sad:

So I need hint with this cursed problem, to be more preciously with my solution, why my solution sux. My solution can give a good approximation with small numbers of D, and L

Actually solution:

I have draw a few coordinate Axis, Theta is perpendicular to lines on the floor, Eta is parallel to them. The endpoints of a Needle, have its own coordinates.

Then I have represented all possible values of coordinates of this endpoints as side of the rectangle. This coordinates of endpoints must be no more than L. From this stuff I have build the equation for the line in the Cartesian Coords. Then I have calculate the total area of my shape, and then I have calculated the area of triangle in which the endpoints in the different "sides" of the line, which is mean the line must be crossed.

Then I have divide this triangle are by the total area, and multiplied by factor two, because there is a two line that can be crossed, and I have obtain the probability that the frigging needle will cross the frigging line. p(U). In case of D = 16, and L = 12, my formula gives probability approx. 0,45, and formula which has been discovered by this mr Buffon, gives to us 0,477.
Perhaps I need little bit correct my stuff, and then it will sux no more? :( .

Thanx for the answers.

P.S. My solution is actually on the images so I hope, this forum will show ma' images, if not - https://s25.postimg.org/bcx5x9ifj/Buffon_stick.jpg, this is a direct link. And sorry for ma' english level.

 

[Buzz Bloom]

Hi frostysh:
I apologize for not referring to your diagram, but I am unable to understand how the diagram represents your work towards a solution.

I offer a suggestion about an approach to the problem. There are two variables related to whether the needle will cross a line.
1. P = the position of the center of the needle with respect to the lines. This variable has a uniform distribution between 0 and L/2.
2. A = the angle of the needle with respect to the lines. This variable has a uniform distribution between 0 and π/2.
You need to calculate the range of values of P and A corresponding to the needle crossing a line.

Regards,
Buzz

 

[Stephen Tashi]

why my solution sux.

In your diagram it is true that all line segments of length L with one endpoint (x,y) that lies in the triangle (D-L, D),(D,D),(D,D+L) is a line that intersects the vertical lines. However, there are also some line segments of length L with one endpoint (x,y) in the figure bounded by (0,0),(D,0),(D,D)(D-L,D)(0,L) that intersect the vertical lines.

And sorry for ma' english level.

Your English would be better if you used ordinary words like "my" instead of slang like "ma". The general impression that slang like "sux", "friggin", "thanx" gives on a USA English speaking forum is that the writer is a teenager or pre-teenager. Of course, that might be correct in your case, but you will taken more seriously if you don't use such slang.

 

[frostysh]

First off - thanks to all, this forum is useful for me indeed.

@Buzz Bloom

Well, it's ok - often even frostysh can't understand the nonsense that frostysh draws on the diagrams ;) . I have spent few month (of course with a long pauses) in my attempts to solve this ... problem. Then I have found the solution that we can see on the diagram, but after I have checked my numeric answer with the "book's" formula, I have realized that my solution little bit s... not so good as it need to be, wel in the small numbers it very close to the right answer, when numeric values of D, and L is increasing, the abyss between my solution and normal is increasing too :/, I am was very sad, so I have look into a books solution, which is including some crazy Integral stuff (that based on the position of the center, and angle - right what you saying), etc, I have understand almost nothin - so I decided to "update my solution" to obtain same answer like in the book :) . But I will think about those integralic stuff too.

@Stephen Tashi

I have no realized why our bloody stick will intersect those lines, on the floor, when values of it's coords will be in the next figure (0,0),(D,0),(D,D)(D-L,D)(0,L), coz' boz of endpoints will be inside of our "belt", perhaps you did mentioned second line, but to count it I have multiplied $p(U)$ by factor two, and that because I have obtain $p(U) \sim L^{2}$, but not $p(U) \sim \frac{1}{2} L^{2}$. I need to think... *thinking*

P.S. About a language - Or, my intellect just not "evolve", and it in the same state as when I was a kiddo, lol... This is bad coz' it's my favorite words: "nonsense", "sux", "frigging", etc. It will be hard to rid from this habits to suse them, but I will try...