What is the Probability of Non-Overlapping Coins on a Rectangular Carpet?

[Physics_wiz]

I remmeber reading before in a probability book that this is an unsolved problem (I don't know if it still is or not):

Given a rectangular carpet with dimensions m x n, find the probability that a coin with radius r will fall on the carpet without touching another coin that's already on the carpet (number of coins on the carpet would have to be known I guess).

 

[iStealth]

It reminded me about the Buffon's Needle Problem: "Buffon's needle problem asks to find the probability that a needle of length l will land on a line, given a floor with equally spaced parallel lines a distance d apart. The problem was first posed by the French naturalist Buffon Eric Weisstein's World of Biography in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104)." (http://mathworld.wolfram.com/BuffonsNeedleProblem.html) .

 

[Physics_wiz]

I found the book where I read that problem. Here's what the problem is:

"We have a rectangular carpet and an indefinete supply of perfect pennies. What is the probability that if we drop the pennies on the carpet at random no two of them will overlap?"

The book says this one hasn't been solved yet, but gives the answer to the one dimensional problem: "Drop n needles of length h on a stick of length b at random. What's the probability that no two needles overlap?"

The answer is [(b-nh)/(b-h)]^n if b>=nh or 0 if b<nh

Any ideas about the first problem? I was thinking: if the one dimensional answer is given, then why not replace each penny by a cross and find the probability that the vertical line will have the same y coordinate as another vertical line then find the probability that the horizontal line will have the same x coordinate as another then multiply them together?

 

[kfmfe04]

Any ideas about the first problem? I was thinking: if the one dimensional answer is given, then why not replace each penny by a cross and find the probability that the vertical line will have the same y coordinate as another vertical line then find the probability that the horizontal line will have the same x coordinate as another then multiply them together?

A different, simpler approach may be this:
- replace all coins with the center of the coin
- find the distance D_ij between all pairs of n points, so i:1 to n, j:1 to n, but j not equal to i
- what is the probability that all D_ij>2r (no overlaps)

It seems to me that this should be a solvable problem, if I integrate like the approach to Buffon's Needle.

Is there some feature I am missing that makes it unsolvable?

 

[SW VandeCarr]

A different, simpler approach may be this:
- replace all coins with the center of the coin
- find the distance D_ij between all pairs of n points, so i:1 to n, j:1 to n, but j not equal to i
- what is the probability that all D_ij>2r (no overlaps)

It seems to me that this should be a solvable problem, if I integrate like the approach to Buffon's Needle.

Is there some feature I am missing that makes it unsolvable?

I don't know but I think at some point you need to calculate every possible arrangement of n coins on the surface available that satisfies your conditions. Of course, the problem doesn't specify how big the carpet is or how small the coins are. Therefore, choose a very small carpet and very big coins. (Actually you want to calculate the probability that coins will be in contact and subtract that from one. Also, the probability is always conditional on the state of the event space at time t(i)).