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'Coin Lines' printed from http://nrich.maths.org/
The person who sent in this solution didn't
add their name, but it's a nice visualisation.
To start off, consider the situation where the distance between the
two lines is double the diameter of the coin. Call the diameter of
the coin d - so we can say the centre of the coin is 0.5d from the
edge of the coin. The distance between the lines is 2d. For the
coin not to cross either line, the centre of the circle must be a
perpendicular distance of between 0.5d and 1.5d (inclusive) away
from a line. The area between the two lines then "allowed" for the
centre of the coin to land in without touching a line is therefore
half of the total area between the two lines - so assuming the
centre of the coin is equally likely to land in all areas, the coin
will touch a line half the time.
Now a good way to think about the concentric circles is to imagine
the coin somewhere and focus on a line through the centre of the
coin and the centre of the concentric circles.
The situation is the same as the one with straight lines I have
considered already. So the answer is 0.5 or d, if the gap is 1 and
the coin diameter is d.