jmc

To count the number of numbers in this set, we subtract 49 from all of the numbers, giving the set $\{1,2,3,\dots ,950\}$, making it obvious that there are 950 numbers total. Furthermore, the set $\{50,51,52,\dots ,98,99\}$ corresponds to the more easily counted $\{1,2,3,\dots ,49,50\}$ by subtracting 49. So, the probability of selecting a two-digit number is $\frac{50}{950}=\boxed{\frac{1}{19}}$.