\(P=\left[1-\frac{1}{2}\right]\left[1-\frac{1}{3}\right]\left[1-\frac{1}{4}\right]...\left[1-\frac{1}{99}\right]\)
\(=\left[\frac{2}{2}-\frac{1}{2}\right]\left[\frac{3}{3}-\frac{1}{3}\right]\left[\frac{4}{4}-\frac{1}{4}\right]...\left[\frac{99}{99}-\frac{1}{99}\right]\)
\(=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot...\cdot\frac{98}{99}\)
\(=\frac{1\cdot2\cdot3\cdot...\cdot98}{2\cdot3\cdot4\cdot...\cdot99}=\frac{1}{99}\)