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.

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.

- 1 month ago

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

- 1 month ago

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