A checkerboard measures 12 inches by 12 inches. There are 8 squares to a side and the checker measures 1 inch in diameter. What is the probability that if someone threw a checker onto the board that it would not be touching a line?
Let each square is L" on a side, let the diameter if the checker be D. If you consider the area in a L" square where if the checker landed it would not be crossing or touching a boundary it would be a square whose side would be L-D. If the checker landed any where inside this square it would not touch a line. To picture this make a sketch of two concentric squares the outer one with side L and the inner one with side L-D.
So the probability is the ratio of the two areas p(not land on a line) =(L-D)2/L2 L=1.5" and D=1" so p=(.5/1.5)2 =.1111 or about 11.1%.