uniform distribution (discrete and real)

Two questions about uniformly distributed variables:

1. Let X be a random real number between 0 and 1. What is the probablity of X's decimal notation containing 123 anywhere? Intuitively I feel that this probability should be 0, because I think there are infinitely more numbers in (0,1) *without* 123 in their decimal notation.

2. Let X be a random natural number. What is the probability of X being even? Intuitively I feel that this probability should be 1/2, because there are just as many odd as even natural numbers.

I'm having a hard time proving either 'officially'..?