 # Buffon's Needle: Length Question

The problem is nicely explained here.

BUT, what happens if the needle lies perpendicular to the lines and entirely covers the space between them? Is it then considered to have crossed one line? Two lines? No lines? Does this go back to some definition of “length” that I should’ve learned (and remembered) from junior high?

Yeah, asking a math/statistics question on a Saturday afternoon. Watch this thread sink like a lead brick.

That may seem like a big but that you have there, Earthling, but it’s not so bad, really. Since the chances that the needle will fall exactly between the lines like that is so small, it doesn’t make much difference how you score it–it’s effect on the final statistics will be down in the “noise.”

In other words, you’re free to come up with whatever convention you want–the probability is still the same.

Remember we’re dealing with mathematical idealisations, that is, the needle has no width, is exactly the distance between the lines, etc.

The probability of the needle being exactly perpendicular to the line is zero (I won’t get technical, but the chance of it being at any specific angle must be the same, and if this was greater than zero then the total probability must be greater than one (indeed, infinite) which is nonsense.)

This means that the probability that the needle crosses the line is unaffected by this case, whatever result you ascribe to it. I personally would say it touches a line, given that it touches one, and the options are ‘touches’ and ‘doesn’t touch’.

I can go into the maths of it in more detail if you’d like.

To give an analagous example, but one which your intuition may work for, imagine we choose a number x (uniformly) randomly between 0 and 1 (inclusive). If 0<x<=0.5 say y=0. If 0.5<x<=1 say y=1. If x=0 then y 0, or (alternatively 1, or 2). The probability of y being 0 is then a half, regardless of what answer we picked for when x was 0. This isn’t rigorous by any means, but I hope you believe me Please do, actually. Hey, lead bricks float! Why does this random drop line crossing probability approximate Pi so closely? Be gentle.

astro, without going in to any details, here’s a quick observation that may help you understand how Pi can pop in there. The landing position of the needle is completely determined by two things: 1. Where the center of the needle falls, and 2. The orientation of the needle.

Pick a particular point at random, and let’s assume that the center of the needle has landed at that point. We don’t know what the needle’s orientation might be–what are the possibilities? Well, fixing the center of the needle, and allowing it to rotate through all possible orientations, we trace out a circle in the plane. And wherever you have a circle, it’s not surprising that Pi should pop up in the details.