Since the lines are parallel you can rephrase the problem:<p>A circle of radius L is centered x far away from a border 0&lt;x&lt;L. This is because one end of the needle will always end in some zone (the center) and the other end will be L far away (the circle).<p>How much of the 2<i>pi boundary is outside the zone?<p>When x -&gt; 0 then it&#x27;s going to be 50% since one boundary line becomes a tangent and the other goes through the middle. When we move x by k (e.g. f(x+k)) then 2</i>k new points will be added on the left side while 2*k points leave the boundary on the right side. When x=L&#x2F;2 then the boundary lines will split the circle in four equal parts (since they&#x27;re tangent to the radius at r&#x2F;2 on both sides) so intuitively its 50%.