What is the probability that coin will not touch lines of segments?

Question

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)

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!

Answer 1

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: $$\frac{2a-2r}{2a}=1-\frac ra.$$

Answer 2

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 $${2a-2r\over2a}.$$