Help: Reframing Buffon's Needle Problem

Hello, It was on 14th March on Pi day that I got to know about Buffon's needle problem:

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

It was worth noticing that how π can come up in unexpected situations. I saw the solution and it was quite easy and straightforward to understand.

By taking \(d_1\) as the distance from the nearest line and \(θ\) as the acute angle between the needle and the projected line with length \(d_2\). We integrated the 2 variable and the probability was \(\boxed{\frac{2L}{πT}}\) where \(L\) is needle length and \(T\) is the equal distance between the strips and \(L < T\).

Now, I reframed the question as :

Suppose we have a floor made of parallel as well as perpendicular strips of wood , each the same width, forming squares and we drop a needle onto the floor (needle is shorter than the width). What is the probability that the needle will lie across a line between two strips?

I tried by taking \(d_{2}\) as the horizontal distance and tried to calculate it by taking triple integration of those 3 variables but I am unable to approach it.

Please help me to approach this problem.

Note by Mohd. Hamza
1 month, 1 week ago

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You may want to refer to Buffon's needle problem and this.

Chew-Seong Cheong - 1 month ago

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Thanks, so the reframed problem is actually Laplace-Buffon Needle Problem.

Mohd. Hamza - 1 month ago

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