[tex]P=\displaystyle\prod_{k=2}^{50}\left(1-\frac1k\right)[/tex]
Write the [tex]k[/tex]-th number in the product as
[tex]1-\dfrac1k=\dfrac kk-\dfrac1k=\dfrac{k-1}k[/tex]
Then
[tex]P=\dfrac{2-1}2\cdot\dfrac{3-1}3\cdot\dfrac{4-1}4\cdot\cdots\cdot\dfrac{49-1}{49}\cdot\dfrac{50-1}{50}[/tex]
[tex]P=\dfrac12\cdot\dfrac23\cdot\dfrac34\cdot\cdots\cdot\dfrac{48}{49}\cdot\dfrac{49}{50}[/tex]
Notice how the denominator of one term cancels with the numerator of the next, leaving us with
[tex]\boxed{P=\dfrac1{50}}[/tex]
