[tex]P = \Big(1-\dfrac{1}{2}\Big)\cdot \Big(1-\dfrac{1}{3}\Big)\cdot\Big(1-\dfrac{1}{4}\Big)\cdot ...\cdot \Big(1-\dfrac{1}{50}\Big) \\ \\ P = \prod\limits_{k=2}^{50}\Big(1-\dfrac{1}{k}\Big) \\ P = \prod\limits_{k=2}^{50}\dfrac{k-1}{k} \\ P = \dfrac{\prod\limits_{k=2}^{50}(k-1)}{\prod\limits_{k=2}^{50}k} \\ P = \dfrac{\prod\limits_{k=2}^{50}(k-1)}{\prod\limits_{k=3}^{50}(k-1)\cdot 50} \\ \\ P = \dfrac{(2-1)\cdot \prod\limits_{k=3}^{50}(k-1)}{\prod\limits_{k=3}^{50}(k-1)\cdot 50}[/tex]
[tex] \\ \\ P = \dfrac{(2-1)}{50} \Rightarrow \boxed{\boxed{P = \dfrac{1}{50}}}[/tex]
