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.