\(P=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}}{1+\dfrac{2010}{2}+1+\dfrac{2009}{3}+...+1+\dfrac{1}{2011}+1}\)
\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}}{\dfrac{2012}{2}+\dfrac{2012}{3}+...+\dfrac{2012}{2011}+\dfrac{2012}{2012}}=\dfrac{1}{2012}\)