Let y be the distance of the lowest point of the needle to the nearest line above it. If the needle falls horizontally, then y will be the distance from the needle to the nearest line above it. So, . Let represent the angle between the needle and the positive direction of x-axis, so that . Then the ordered pair determines the position of the needle.
So the square: forms the sample space. The quantity will be the the vertical height of the needle. Now, we know that the needle will intersect any one of the lines if and only if . So the event space(favourable) is given by the area of region that lies between the curves and . Thus the required probability is equal to :
In our question , and
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. Let 
represent the angle between the needle and the positive direction of x-axis, so that 
. Then the ordered pair 
determines the position of the needle.
forms the sample space. The quantity 
will be the the vertical height of the needle. Now, we know that the needle will intersect any one of the lines if and only if 
. So the event space(favourable) is given by the area of region that lies between the curves 
and 
. Thus the required probability is equal to : 
and 
