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In geometric probability I want to know what is the probability that a small coin (r < a) will not touch the lines of segments with 2*a distance(I mean the plain is divided into 2*a length segments with parallel lines and the coin should not touch this lines)

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from each other. I thought it would be 2r/2a but this is not the correct answer. Can you help, please? Thanks for your time!

  • Consider how much of the ground is acceptable for the centre of the coin to land on. Also see "Buffon's needle" as this seems like some kind of prepatory question about that. – Leonhard Euler Oct 25 '18 at 14:08
  • I'm not a specialist on this, but $2r/2a$ can't be right because the probability would get smaller when $r$ decreases, which is obviously wrong. If the center of the coin is between $r$ and $2a - r$ then the coin doesn't touch the lines. So I'd expect a probability like $\frac{2a -2r}{2a}$ – Ronald Oct 25 '18 at 14:08

2 Answers2

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Consider the circle $x^2+(y-t)^2=r^2$ with the center $(0,t)$ and the radius $r$. Also consider the line $y=2a$. The circle will be within the lines $y=0$ and $y=2a$ when: $$r<t<2a-r.$$ Hence the probability is: $$\frac{2a-2r}{2a}=1-\frac ra.$$

farruhota
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If the center of the coin is within $r$ of the line, then then coin touches the line. There are two "forbidden strips" of width $r$ in each strip of width $2a,$ so the probability is $${2a-2r\over2a}.$$

saulspatz
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